Minors for alternating dimaps

Minors for alternating dimaps Abstract We develop a theory of minors for alternating dimaps—orientably embedded digraphs where, at each vertex, the incident edges (taken in the order given by the embedding) are directed alternately into, and out of, the vertex. We show that they are related by the triality relation of Tutte. They do not commute in general, though do in many circumstances, and we characterize the situations where they do. We give a characterization of alternating dimaps of at most a given genus, using a finite set of excluded minors. We also use the minor operations to define simple Tutte invariants for alternating dimaps and characterize them. We establish a connection with the Tutte polynomial, and pose the problem of characterizing universal Tutte-like invariants for alternating dimaps based on these minor operations. 1. Introduction The minor relation is one of the most important order relations on graphs. A graph H is a minor of a graph G if it can be obtained from G by some sequence of deletions and contractions of edges. Many important classes of graphs can be characterized by the exclusion of some finite set of minors. These include forests, series-parallel graphs [15, 19], outerplanar graphs [12], planar graphs [30, 40]—and, in fact, any minor-closed class of graphs, by Robertson and Seymour’s proof of Wagner’s conjecture [35]. Minors also play a central role in enumerative graph theory: the Tutte–Whitney polynomials, which contain information on a great variety of counting problems on graphs or matroids, satisfy recurrence relations using deletion and contraction (see, for example [9, 21, 24, 41]). The theory of minors derives much of its richness and beauty from the fact that the deletion and contraction operations are dual (in the sense of planar graph duality or, more generally, matroid duality [34]) and commute. In this paper, we introduce and study a minor relation on alternating dimaps. An alternating dimap is a directed graph without isolated vertices, 2-cell-embedded in a disjoint union of orientable 2-manifolds, where each vertex has even degree and, for each vertex v, the edges incident with v are directed alternately into, and out of, v (when considered in the order in which they appear around v in the embedding). An alternating dimap may have loops and/or multiple edges, but cannot have a bridge. We allow the empty alternating dimap with no vertices, edges or faces. For alternating dimaps, we have three minor operations, instead of two. We show in Section 2 that they are related by a triality relation of Tutte [37], in a manner analogous to the duality between deletion and contraction. The form of the relationship is the same as that found by the author for some other combinatorial objects (binary functions) on which minors and triality can be defined [26]. One property of ordinary minor operations (and also of the minor operations in [26]) is that they commute. We show in Section 3 that minor operations on alternating dimaps do not commute in general, although they do in most circumstances, and we determine exactly when they do. As seen in the first paragraph, two of the main themes of the classical theory of minors are excluded minor characterizations and Tutte invariants. The remainder of this paper takes these themes up for alternating dimaps and their minor operations. In Section 4, we give an excluded minor characterization of alternating dimaps of at most a given genus, using a finite set of excluded minors in every case. In Section 5, we define simple Tutte invariants for alternating dimaps, and show that there are only a few of them and that they do not contain much information, in contrast to the situation for graphs, matroids and binary functions. We then define extended Tutte invariants and raise the question of how many and varied they might be. We show that they are much richer than the simple Tutte invariants, as they include, in a sense, the Tutte polynomial of a planar graph. Sections 3–5 may be read independently of each other. 1.1. History and related work Plane alternating dimaps were studied by Tutte [37, 38]. He showed that they come in triples, with the three members of a triple being derivable from a single larger structure, a bicubic map (see below). The relationship among the members of such a triple is called triality [37] or trinity [38], and each is trial or trine to the others. This relationship extends ordinary duality. Tutte’s original motivation was to determine when equilateral triangles can be tessellated by smaller equilateral triangles of different sizes or orientations. He also proved his Tree Trinity Theorem, on spanning arborescences of such maps. In the second paper [38], he noted the possibility of extending this theory to other surfaces. Berman [2] showed explicitly how to construct the trial of an alternating dimap without reference to the bicubic map from which three trial maps are derived, and gave alternative proofs of some of Tutte’s results. Tutte reviewed some aspects of his theory in [39]. Tutte showed that a ‘triangulated triangle’ gives rise to a plane bipartite cubic graph (a plane bicubic map), which in turn has, as its dual, an Eulerian plane triangulation. The triple of trial alternating dimaps is derived from this bicubic map. Although bicubic maps are plane in Tutte’s work, they may more generally be taken to be embedded in some orientable surface, so each has a genus. It is interesting to note that this stream of research, first seen in Tutte’s 1948 paper [37], can be traced back to the same source that eventually gave rise to Tutte’s work on minor operations and his eponymous polynomial. Historically, the source of both streams was the famous paper on ‘squaring the square’ [7]. The 1948 paper extended the theory to ‘triangulating the triangle’ (where all triangles are equilateral) and introduced triality, among other things. However, this stream has not previously seen the development of minor operations or Tutte-like invariants for alternating dimaps. In one of their later and lesser-known papers, Brooks et al. introduced reductions for plane bicubic maps that correspond to ours [8, Section 11], although they are not translated into alternating dimap form, are not treated as minor operations, and are used for a different purpose. Jaeger [29] used related reductions on plane bicubic maps. A separate stream of research concerns latin bitrades, which are pairs of partial latin squares of the same shape and with the same symbol set in each row and column. These may also be given a genus. Cavenagh and Lisoněk [10] established a correspondence between spherical latin bitrades and 3-connected planar Eulerian triangulations (dually, 3-connected plane bicubic maps), while the relationship between spherical latin bitrades and triangulated triangles is described in [16, 17] (see also [11, 18]). Batagelj [1] introduced two operations on plane Eulerian triangulations by means of which larger such maps can be generated from smaller ones. These operations have been used and extended in several papers (via the aforementioned correspondences) to generate latin bitrades [17, 28]. The inverse of one of these operations, translated to alternating dimaps, corresponds to (technically, restricted versions of) our minor operations. (I thank Tony Grubman and Ian Wanless for pointing out this link.) Although duality, with associated minors, appears in many forms for many different kinds of objects, there are far fewer settings with natural minor operations related by triality. The main ones known to the author are alternating dimaps, binary functions [26] (see also [22, 23, 24, 25]), multimatroids (including isotropic systems) [5, 6] and the related transition matroids [36, pp. 8, 10]. Some alternating dimaps certainly cannot be represented by objects of these other types, since alternating dimap minor operations may not commute, unlike those in the other settings. In [27, Section 4], we determine those alternating dimaps that can be represented faithfully by binary functions. 1.2. Definitions and notation If G is an alternating dimap, then kG is the disjoint union of k copies of G. Each of these copies is regarded as being embedded in a different surface, with all these k surfaces being disjoint from each other. An edge e from u to v is sometimes written e(u,v). Let G be an alternating dimap, viewed topologically as embedded in an orientable surface. Let C⊆E(G) be a circuit of G (which need not have all its edges directed the same way), and let S be the connected surface in which the component of G containing C is embedded. Then the sides of C are the components of S−C. A side is planar if it is homeomorphic to the open unit disc. The genus γ(G) of an alternating dimap G is given by   ∣V(G)∣−∣E(G)∣+∣F(G)∣=2(k(G)−γ(G)),where F(G) is the set of faces of G and k(G) is the number of components of G. The edges around a face all go in the same direction, and we say the face is clockwise or anticlockwise according to the direction of the edges around it. We identify a clockwise (respectively, anticlockwise) face with its cyclic sequence of edges, and call it a c-face (respectively, a-face) for short. Observe that the (edge sets of the) c-faces partition E(G), as do the a-faces. So every edge e belongs to one c-face, denoted by C(e)=CG(e), and one a-face, denoted by A(e)=AG(e). If two faces share a common edge, then one of the faces is clockwise and the other is anticlockwise. The left successor (respectively, right successor) of an edge e is the next edge along from e, going around AG(e) (respectively, CG(e)) in the direction given by e. (This direction is anticlockwise for the left successor, and clockwise for the right successor.) Often, c-faces and a-faces are simple cycles, but this is not always the case. If v is a cutvertex of G, then one face incident with v consists of two or more edge-disjoint cycles. The numbers of clockwise and anticlockwise faces of G are denoted by cf(G) and af(G), respectively. An in-star is the set of all edges directed into some vertex. So in-stars are in one-to-one correspondence with vertices. The in-star of edges directed into vertex v is denoted by I(v)=IG(v). Observe that the in-stars partition E(G), so every edge e also belongs to one in-star, denoted by I(e)=IG(e) (overloading notation slightly). If an alternating dimap is disconnected, then we treat its components as being embedded in separate, disjoint surfaces. Alternating dimaps extend ordinary embedded graphs (in orientable surfaces), in that replacing each edge of an embedded graph by a pair of directed edges, forming a clockwise face of size 2, gives an alternating dimap [38]. An alternating dimap G defines three permutations σG,1,σG,ω,σG,ω2:E(G)→E(G) (abbreviated σ1,σω,σω2 where G is clear from the context), as follows. For each e∈E(G), its image under σG,1, σG,ω and σG,ω2 is the next edge in clockwise order around IG(e), AG(e) and CG(e), respectively. So the left successor of e is σG,ω−1(e), while the right successor of e is σG,ω2(e). Note that, in going around an in-star in clockwise order, we skip outgoing edges at the vertex as these do not belong to the in-star. While ω can often be treated just as a symbol, at times we do algebra with it, using the value ω=exp(2πi/3). Any two of our three permutations determine the other. This follows from the relation σ1σωσω2=idE(G), the identity permutation on E(G) (with permutations applied from right to left). Let E be any finite set and let SE be the set of all triples (σ0,σ1,σ2) of permutations, each acting on E, such that σ0σ1σ2=idE. This is called a 3-constellation or a hypermap [31]. When one of the permutations is an involution, we may take its cycles to correspond to undirected edges (using the aforementioned representation of embedded graphs by alternating dimaps), and we have a standard combinatorial representation of an orientably embedded graph (see, for example [4, Section 2.2]). In the general case, we have an equivalence with alternating dimaps on E (i.e., whose edges are labelled by E) which seems to be well known (see for example [13]) although I have not seen it stated explicitly. Proposition 1.1 The map {alternatingdimapsonE}→SEgiven by G↦(σG,1,σG,ω,σG,ω2)is a bijection.□ If G is an alternating dimap, then the trial Gω of G is defined as follows. Its vertices represent the c-faces of G. We denote the vertex of Gω representing c-face C by vC. (Think of vC being placed inside C in the embedding.) Edges of Gω are constructed as follows. Suppose two c-faces C1 and C2 of G share a vertex v, and that there is an a-face A containing edges e and f going into and out of v, respectively, with e and f also belonging to C1 and C2, respectively. (See Fig. 1(a). We do not require e and f to be distinct, or C1 and C2 to be distinct.) Then we put an edge eω from vC2 to vC1 in Gω. Figure 1. View largeDownload slide Construction of trial dimap Gω from G: (a) clockwise faces → vertices and e↦eω; (b) example showing G (solid edges), Gω (dashed edges) and Gω2 (dotted edges). Note that Gω3=G. Figure 1. View largeDownload slide Construction of trial dimap Gω from G: (a) clockwise faces → vertices and e↦eω; (b) example showing G (solid edges), Gω (dashed edges) and Gω2 (dotted edges). Note that Gω3=G. These edges of Gω are ordered around C1 according to the order of the edges e around C1. Similarly, they are ordered around C2 according to the order of the edges f around C2. (Think of eω as being drawn by a curve from vC2, inside C2, to its destination vC1 inside C1, in such a way that it crosses f in its ‘first half’ (i.e., closer to its start than its end) and crosses e in its ‘last half’.) An example of the construction, for an alternating dimap on three edges, is given in Fig. 1(b). It is routine to show that the map •ω:E(G)→E(Gω), e↦eω is a bijection, and that the c-faces, a-faces and in-stars of Gω are the a-faces, in-stars and c-faces, respectively, of G. We can also express this relationship in the language of the permutation triples. The first permutation σGω,1 in the permutation triple for Gω represents (by its cycles) the in-stars of Gω. These correspond to c-faces of G, which are represented by (the cycles of) σG,ω2. Proposition 1.2   σG,1(e)ω=σGω,ω(eω) (1.1)  σG,ω(e)ω=σGω,ω2(eω) (1.2)  σG,ω2(e)ω=σGω,1(eω). (1.3)□ If G is represented by (σG,1,σG,ω,σG,ω2), then its trial Gω is represented by (σG,ω2,σG,1,σG,ω). It is clear that Proposition 1.2 still holds if σ is replaced by σ−1 throughout. The trial operation on a component of G is independent of the other components. We write Gω2 for (Gω)ω. From the way triality changes c-faces to in-stars to a-faces to c-faces, we find that Gω3=(Gω2)ω=G1=G. 1.3. Loops and semiloops A (standard) loop is just an edge of G which is a loop in the undirected version of G. If it is a separating circuit of the embedding, then it divides its component of the embedding surface into two sides, its clockwise side and its anticlockwise side. If it is non-separating, then it has just one side, which we take to be both its clockwise and anticlockwise side. A 1-loop is an edge whose head has degree 2. This does not need to be a standard loop, since its two vertices need not coincide. It is an edge whose left and right successors are identical, and, in such cases, we can refer unambiguously to its successor. An ω-loop is an edge forming a single-edge a-face. An ω2-loop is an edge forming a single-edge c-face. Unlike 1-loops, ω-loops and ω2-loops are standard loops. However, not every standard loop is of this type, as we will see when considering 1-semiloops shortly. For any alternating dimap G, an edge e is a 1-loop in G if and only if eω is an ω-loop in Gω, which in turn holds if and only if eω2 is an ω2-loop in Gω2. A triloop is an edge which is a μ-loop for some μ∈{1,ω,ω2}. An ultraloop is a triloop which (together with its vertex) constitutes a component of the graph. It has two faces, a c-face and an a-face, and its vertex has degree 2, so it is simultaneously a 1-loop, an ω-loop and an ω2-loop. In fact, if an edge is a μ-loop for μ equal to any two of {1,ω,ω2}, then it is an ultraloop. A 1-loop is only a standard loop if it is an ultraloop. A μ-loop is a proper μ-loop, and a proper triloop, if it is not also an ultraloop. In such a case, it is not a ν-loop for any ν∈{1,ω,ω2}⧹{μ}. A 1-semiloop is just a standard loop. An ω-semiloop is an ω2-loop or an edge e such that deleting both e and its right successor σω2(e) either increases the number of components of G or decreases its genus. (After deletion, we no longer have an alternating dimap in general; we are really referring to the underlying undirected embedded graph here, rather than G itself.) This latter condition may be written: k(G⧹{e,σω2(e)})−γ(G⧹{e,σω2(e)})>k(G)−γ(G). Similarly, an ω2-semiloop is an ω-loop or an edge e such that deleting both e and its left successor σω−1(e) either increases the number of components of G or decreases its genus. For each μ∈{1,ω,ω2}, a proper μ-semiloop is a μ-semiloop that is not a triloop. A proper 1-semiloop e either gives a non-contractible closed curve in the embedding, or each of its two sides contains an edge other than e from the same component as e. An edge is a μ-semiloop in G if and only if it is a μω-semiloop in Gω. If μ1≠μ2, an edge is both a μ1-semiloop and a μ2-semiloop if and only if it is a (μ1μ2)−1-loop. The effect of the trial construction on each type of loop can be seen in Fig. 1(b), where the initial dimap G contains an ω2-loop, a 1-loop and an ω-semiloop (each proper). 2. Minors Let G be an alternating dimap and e=e(u,v)∈E(G). If e is not a loop, then the dimap G[1]e is formed by deleting the edge e and identifying its endpoints, while preserving the order of the edges and faces around vertices. If e is an ω-loop or an ω2-loop, then G[1]e is formed just by deleting e. If e is a 1-semiloop, then G[1]e is formed as follows. Let the edges incident with v, in cyclic clockwise order around v starting with e directed into v, be e,a1,b1,…,ak,bk,e,c1,d1,…,cl,dl. Here, each ai and each di is directed out of v, while each bi and ci is directed into v. We replace v by two new vertices, v1 and v2, and reconnect the edges ai,bi,ci,di as follows. The tail of each ai and the head of each bi becomes v1 instead of v, while the head of each ci and the tail of each di becomes v2 instead of v. The edge e is deleted. The cyclic orderings of edges around v1 and v2 are those induced by the ordering around v. Everything else is unchanged. Observe that if e is a separating 1-semiloop then, in effect, the shrinking of the loop severs the subdimap in its clockwise side from that in its anticlockwise side. Each of these subdimaps is now on a separate surface, so the number of components has increased by 1. If e is not separating then, in effect, shrinking the loop cuts one of the handles of the surface component in which it is embedded, so reducing the genus by 1. This operation is called 1-reduction or contraction. See Fig. 2(a,b). It just adapts the usual contraction operation for surface minors to the alternating dimap context. Let f=f(v,w1) be the right successor of e, and let g=g(v,w2) be the left successor of e. (It is possible that f=g, in which case w1=w2. This occurs when v has indegree=outdegree=1, i.e., when e is a 1-loop.) Figure 2. View largeDownload slide Minor operations: G and reductions (black, solid edges, filled vertices), with their trials, which equal Gω and reductions (dashed edges, open vertices). (a) G and Gω, (b) G[1]e and (G[1]e)ω = Gω[ω2]eω, (c) G[ω]e and (G[ω]e)ω = Gω[1]eω, (d) G[ω2]e and (G[ω2]e)ω = Gω[ω]eω. Figure 2. View largeDownload slide Minor operations: G and reductions (black, solid edges, filled vertices), with their trials, which equal Gω and reductions (dashed edges, open vertices). (a) G and Gω, (b) G[1]e and (G[1]e)ω = Gω[ω2]eω, (c) G[ω]e and (G[ω]e)ω = Gω[1]eω, (d) G[ω2]e and (G[ω2]e)ω = Gω[ω]eω. The graph G[ω]e is formed by deleting e and, if e≠g, changing g so that it now joins u to w2. The revised edge g replaces e and g in A(e), and it replaces g in I(w2). If degv=2, then v and I(v) no longer exist in G[ω]e. If degv≠2, then the in-star at v in G[ω]e is I(v)⧹{e}. The c-face containing g in G[ω]e is (C(e)⧹{e})∪C(g). This operation is called ω-reduction. See Fig. 2(a,c). The graph G[ω2]e is formed by deleting e and, if e≠f, changing f so that it now joins u to w1. The revised edge f replaces e and f in C(e), and it replaces f in I(w1). If degv=2, then v and I(v) no longer exist in G[ω2]e. If degv≠2, then the in-star at v in G[ω2]e is I(v)⧹{e}. The a-face containing f in G[ω2]e is (A(e)⧹{e}∪A(f)). This operation is called ω2-reduction. See Fig. 2(a,d). For all μ∈{1,ω,ω2}, μ-reduction of a proper μ−1-semiloop either increases the number of connected components or decreases the genus. Each of these three operations is a reduction or a minor operation. The operations of ω-reduction and ω2-reduction are special cases of lifting (see, for example [32]), used in the immersion relation on graphs [33], here restricted to cases where the two incident edges are consecutive in a face. An alternating dimap obtained from G by a sequence of minor operations is a minor of G. If e is a triloop, then G[1]e=G[ω]e=G[ω2]e. We sometimes write G[∗]e for the common result of the three reductions in this case, in order to avoid being unnecessarily specific. If X=(x1,…,xk) is a sequence of edges, then we write G[μ]X as a shorthand for G[μ]x1[μ]x2⋯[μ]xk. It is straightforward to translate the above constructions for the minor operations into the language of permutation triples. Theorem 2.1 If G is an alternating dimap with permutation triple (σ1,σω,σω2), then the permutation triples for the three minors of G are as given in the following table. G[1]e  G[ω]e  G[ω2]e  σG[1]e,1(σ1−1(e))=σω2−1(e)σG[1]e,ω(σω−1(e))=σω(e)σG[1]e,ω2(σω2−1(e))=σω2(e)σG[1]e,1(σω(e))=σ1(e)Otherwise:σG[1]e,μ(f)=σμ(f)  σG[ω]e,1(σ1−1(e))=σ1(e)σG[ω]e,ω(σω−1(e))=σω(e)σG[ω]e,ω2(σω2−1(e))=σω−1(e)σG[ω]e,ω2(σ1(e))=σω2(e)σG[ω]e,μ(f)=σμ(f)  σG[ω2]e,1(σ1−1(e))=σ1(e)σG[ω2]e,ω(σω−1(e))=σ1−1(e)σG[ω2]e,ω2(σω2−1(e))=σω2(e)σG[ω2]e,ω(σω2(e))=σω(e)σG[ω2]e,μ(f)=σμ(f)  G[1]e  G[ω]e  G[ω2]e  σG[1]e,1(σ1−1(e))=σω2−1(e)σG[1]e,ω(σω−1(e))=σω(e)σG[1]e,ω2(σω2−1(e))=σω2(e)σG[1]e,1(σω(e))=σ1(e)Otherwise:σG[1]e,μ(f)=σμ(f)  σG[ω]e,1(σ1−1(e))=σ1(e)σG[ω]e,ω(σω−1(e))=σω(e)σG[ω]e,ω2(σω2−1(e))=σω−1(e)σG[ω]e,ω2(σ1(e))=σω2(e)σG[ω]e,μ(f)=σμ(f)  σG[ω2]e,1(σ1−1(e))=σ1(e)σG[ω2]e,ω(σω−1(e))=σ1−1(e)σG[ω2]e,ω2(σω2−1(e))=σω2(e)σG[ω2]e,ω(σω2(e))=σω(e)σG[ω2]e,μ(f)=σμ(f) □ For any reduction and any edge, if none of the first four equations in the appropriate column of the table apply, then the last one in the column is used (where, always, f∈E⧹{e}). Each equation in the table can only be used when the argument and image of the permutation both belong to the edge set of the minor, that is to E⧹{e}. So, if some σμ(e) or σμ−1(e) actually equals e, then that equation does not apply in that case. This can only happen when e is a triloop. We will refer to a specific equation in the table by ‘Theorem 2.1 (r,c)’, where r and c index the row and column in which the equation appears. For example, Theorem 2.1(2,1) is σG[1]e,ω(σω−1(e))=σω(e), and Theorem 2.1(5,2) is σG[ω]e,μ(f)=σμ(f) which holds when the pair μ,f is not covered by any of the previous entries in that column. We are now in a position to establish the relationship between minors and triality. Theorem 2.2 If e∈E(G)and μ,ν∈{1,ω,ω2}then  Gμ[ν]eμ=(G[μν]e)μ. Theorem 2.2 extends the classical relationship between duality and minors:   G*⧹e=(G/e)*,G*/e=(G⧹e)*.The analogy between the two settings is shown in Figs. 3 and 4. The former shows the classical relationship, while the latter illustrates Theorem 2.2. Proof of Theorem 2.2 The case μ=1 is trivial. We begin with μ=ω, and first prove that   Gω[1]eω=(G[ω]e)ω. (2.1) It is straightforward when e is an ultraloop. So suppose that e is not an ultraloop.   σGω[1]eω,1(σGω,1−1(eω))=σGω,ω2−1(eω)(byTheorem2.1(1,1))=σG,ω−1(e)ω(by(1.2))=σG[ω]e,ω2(σG,ω2−1(e))ω(byTheorem2.1(3,2))=σ(G[ω]e)ω,1(σG,ω2−1(e)ω)(by(1.3))=σ(G[ω]e)ω,1(σGω,1−1(eω))(by(1.3)).Similarly, we have   σGω[1]eω,ω(σGω,ω−1(eω))=σGω,ω(eω)=σG,1(e)ω=σG[ω]e,1(σG,1−1(e))ω=σ(G[ω]e)ω,ω(σG,1−1(e)ω)=σ(G[ω]e)ω,ω(σGω,ω−1(eω)),σGω[1]eω,ω2(σGω,ω2−1(eω))=σGω,ω2(eω)=σG,ω(e)ω=σG[ω]e,ω(σG,ω−1(e))ω=σ(G[ω]e)ω,ω2(σG,ω−1(e)ω)=σ(G[ω]e)ω,ω2(σGω,ω2−1(eω)),σGω[1]eω,1(σGω,ω(eω))=σGω,1(eω)=σG,ω2(e)ω=σG[ω]e,ω2(σG,1(e))ω=σ(G[ω]e)ω,1(σG,1(e)ω)=σ(G[ω]e)ω,1(σGω,ω(eω)). It remains to consider the cases where   (μ,fω)∈{(1,σGω,1−1(eω)),(ω,σGω,ω−1(eω)),(ω2,σGω,ω2−1(eω)),(1,σGω,ω(eω))}. (2.2)Equivalently,   (μω2,f)∈{(1,σG,1−1(e)),(ω,σG,ω−1(e)),(ω2,σG,ω2−1(e)),(ω2,σG,1(e))}. (2.3)In these cases, we have   σGω[1]eω,μ(fω)=σGω,μ(fω)(byTheorem2.1(5,1),using(2.2))=σG,μω2(f)ω(by(1.1)–(1.3))=σG[ω]e,μω2(f)ω(byTheorem2.1(5,2),using(2.3))=σ(G[ω]e)ω,μ(fω)(by(1.1)–(1.3)). We have now shown that, for all μ∈{1,ω,ω2} and f∈E(G),   σGω[1]eω,μ(fω)=σ(G[ω]e)ω,μ(fω).It follows that Gω[1]eω and (G[ω]e)ω have the same permutation triple and so are equal. Similar arguments show that   Gω[ω]eω=(G[ω2]e)ω,Gω[ω2]eω=(G[1]e)ω.From these, it follows that   Gω2[1]eω2=(G[ω2]e)ω2,Gω2[ω]eω2=(G[1]e)ω2,Gω2[ω2]eω2=(G[ω]e)ω2.□ Figure 3. View largeDownload slide The relationship between the ordinary duality and minor operations. Figure 3. View largeDownload slide The relationship between the ordinary duality and minor operations. Figure 4. View largeDownload slide The relationship between triality and the three minor operations. The diagram wraps around at its left and right sides. Arrows downwards from G,Gω,Gω2 are minor operations; the other arrows (all rightward) are triality. Figure 4. View largeDownload slide The relationship between triality and the three minor operations. The diagram wraps around at its left and right sides. Arrows downwards from G,Gω,Gω2 are minor operations; the other arrows (all rightward) are triality. 3. Non-commutativity Deletion and contraction are well known to commute, in the sense that, for any graph G and any distinct e,f∈E(G), we have   G⧹e⧹f=G⧹f⧹e,G/e/f=G/f/e,G⧹e/f=G/f⧹e.The variants of these operations for embedded graphs, where deletion/contraction of an edge is accompanied by appropriate modifications to the embedding, also commute (see for example [3, Section 2.3]). Perhaps surprisingly, the reductions we have introduced for alternating dimaps do not always commute. Figure 5 illustrates the fact that, in general, if f=σG,ω(e), then G[1]e[ω]f≠G[ω]f[1]e. By triality, it follows that if f=σG,ω2(e), then in general G[ω2]e[1]f≠G[1]f[ω2]e, and if f=σG,1(e), then in general G[ω]e[ω2]f≠G[ω2]f[ω]e. Figure 5. View largeDownload slide Non-commutativity of minor operations. Figure 5. View largeDownload slide Non-commutativity of minor operations. In this section, we prove that these three situations are the only ones in which the reductions do not commute. We first show that two reductions of the same type always commute. Theorem 3.1 For all μ∈{1,ω,ω2},   G[μ]e[μ]f=G[μ]f[μ]e. Proof We show that   G[1]e[1]f=G[1]f[1]e, (3.1)which takes up most of the proof, and then use triality to complete it. To show (3.1), we will show that, for all μ∈{1,ω,ω2} and all g∈E(G)⧹{e,f},   σG[1]e[1]f,μ(g)=σG[1]f[1]e,μ(g). (3.2) We first do this for μ∈{ω,ω2}, which we now assume. Most situations are covered by the following reasoning:   σG[1]e[1]f,μ(g)=σG[1]e,μ(g)ifg≠σG[1]e,μ−1(f)=σG,μ(g)ifg≠σG,μ−1(e)=σG[1]f,μ(g)ifg≠σG,μ−1(f)=σG[1]f[1]e,μ(g)ifg≠σG[1]f,μ−1(e),by four applications of Theorem 2.1(5,1), since the conditions on g ensure that cases (2,1) and (3,1) (according as μ=ω or μ=ω2) of that theorem do not apply. We now deal with situations where the above conditions on g are not met. We have, apparently, four exceptional values of g. We consider each in turn. First, suppose g=σG,μ−1(e). In this case, we must assume that f≠σG,μ−1(e), else g=f and g∈domσG[1]e[1]f,μ. Consider σG[1]e[1]f,μ(g). If f=σG,μ(e), then σG[1]e,μ−1(f)=σG,μ−1(e), by Theorem 2.1(2,1), (3,1). This justifies the first step in the following:   σG[1]e[1]f,μ(σG,μ−1(e))=σG[1]e[1]f,μ(σG[1]e,μ−1(f))=σG[1]e,μ(f)(byTheorem2.1(2,1)or(3,1))=σG,μ(f)(byTheorem2.1(5,1),sincef≠σG,μ−1(e)). On the other hand, if f≠σG,μ(e), then σG[1]e,μ−1(f)=σG,μ−1(f), by Theorem 2.1(5,1). This in turn does not equal σG,μ−1(e), since e≠f and σG,μ−1 is a bijection. So   σG[1]e[1]f,μ(σG,μ−1(e))=σG[1]e,μ(σG,μ−1(e))=σG,μ(e),by Theorem 2.1(5,1), then (2,1) or (3,1). Now consider σG[1]f[1]e,μ(g). Observe that σG[1]f,μ−1(e)=σG,μ−1(e), by Theorem 2.1(5,1), since f≠σG,μ−1(e). Therefore,   σG[1]f[1]e,μ(σG,μ−1(e))=σG[1]f[1]e,μ(σG[1]f,μ−1(e))=σG[1]f,μ(e)(byTheorem2.1(2,1)or(3,1))={σG,μ(f),ife=σG,μ−1(f),σG,μ(e),otherwise,by Theorem 2.1(2,1) or (3,1), and (5,1). So σG[1]e[1]f,μ(g)=σG[1]f[1]e,μ(g) when g=σG,μ−1(e). Secondly, suppose g=σG,μ−1(f). This can be treated the same as the first case, except that e and f are swapped throughout. Thirdly and fourthly, the remaining two exceptional values of g, namely σG[1]e,μ−1(f) and σG[1]f,μ−1(e), are really nothing new, for application of Theorem 2.1 gives   σG[1]e,μ−1(f)={σG,μ−1(e),iff=σG,μ(e),σG,μ−1(f),otherwise;σG[1]f,μ−1(e)={σG,μ−1(f),ife=σG,μ(f),σG,μ−1(e),otherwise.Thus, in any event, each of these two values of g actually falls into one of the first two cases. This completes the treatment of the exceptional values of g (apparently four in number, but really just two). We have now proved (3.2) for μ∈{ω,ω2}. But it then follows immediately for μ=1 too, since σH,1=σH,ω2−1◦σH,ω−1 for any H. So (3.2) holds for all μ and all g, which establishes (3.1). Now that we know contractions commute, we can use triality to show that any two reductions of the same type commute. For any μ∈{1,ω,ω2},   G[μ]e[μ]f=(Gμ[1]eμ[1]fμ)μ−1=(Gμ[1]fμ[1]eμ)μ−1=G[μ]f[μ]e,using Theorem 2.2 twice.□ We next show that the reductions always commute if one of the edges involved is a triloop. Lemma 3.2 If f is a triloop, then for any ν∈{1,ω,ω2},   G[1]e[ν]f=G[ν]f[1]e. Proof If f is a triloop, then any ν-reduction of f just amounts to contraction of f. So   G[1]e[ν]f=G[1]e[1]f=G[1]f[1]e=G[ν]f[1]e,where the middle equality follows from Theorem 3.1.□ Theorem 3.3 If f is a triloop and μ,ν∈{1,ω,ω2}, then  G[μ]e[ν]f=G[ν]f[μ]e. Proof   G[μ]e[ν]f=(Gμ[1]eμ[νμ−1]fμ)μ−1(byTheorem2.2)=(Gμ[νμ−1]fμ[1]eμ)μ−1(byLemma3.2)=G[ν]f[μ]e(byTheorem2.2).□ Most of the remainder of this section is devoted to showing that the two different reductions commute in the remaining situations not covered in Fig. 5 and its two trials. We will need some lemmas. The first uses Theorem 2.1 to describe the inverses of the permutations representing the three minors. Lemma 3.4 (a)   σG[1]e,ω−1(h)={σG,ω−1(e),ifh=σG,ω(e);σG,ω−1(h),otherwise.(b)   σG[ω]f,ω−1(h)={σG,ω−1(f),ifh=σG,ω(f);σG,ω−1(h),otherwise.(c)   σG[1]e,1−1(h)={σG,ω(e),ifh=σG,1(e);σG,1−1(e),ifh=σG,ω2−1(e);σG,1−1(h),otherwise.(d)   σG[ω]f,1−1(h)={σG,1−1(f),ifh=σG,1(f);σG,1−1(h),otherwise.(e)   σG[1]e,ω2−1(h)={σG,ω2−1(e),ifh=σG,ω2(e);σG,ω2−1(h),otherwise.(f)   σG[ω]f,ω2−1(h)={σG,ω2−1(f),ifh=σG,ω−1(f);σG,1(f),ifh=σG,ω2(f);σG,ω2−1(h),otherwise. As in Theorem 2.1, the various expressions must only be applied when h is an edge in the appropriate minor. Proof Immediate from: (a) Theorem 2.1(2,1), (5,1); (b) Theorem 2.1(2,2), (5,2); (c) Theorem 2.1(1,1), (4,1), (5,1); (d) Theorem 2.1(1,2), (5,2); (e) Theorem 2.1(3,1), (5,1); (f) Theorem 2.1(3,2), (4,2), (5,2).□ Lemma 3.5 If f≠σG,ω(e), then  σG[1]e[ω]f,ω=σG[ω]f[1]e,ω. Proof The proof has some similarities to that of Theorem 3.1, but is significantly more complicated. We prove that   σG[1]e[ω]f,ω(g)=σG[ω]f[1]e,ω(g)for all g∈E(G)⧹{e,f}. If g∉{σG,ω−1(e),σG,ω−1(f),σG[1]e,ω−1(f),σG[ω]f,ω−1(e)}, then   σG[1]e[ω]f,ω(g)=σG[1]e,ω(g)=σG,ω(g)=σG[ω]f,ω(g)=σG[ω]f[1]e,ω(g),by Theorem 2.1(5,1), (5,2). This leaves four special cases for g, which we will consider in turn, after noting some facts which we will use repeatedly. Observe that the Lemma’s condition, f≠σG,ω(e), implies   σG[1]e,ω−1(f)≠σG,ω−1(e), (3.3)by Lemma 3.4(a). Also, by Lemma 3.4(b),   e=σG,ω(f)⟺σG[ω]f,ω−1(e)=σG,ω−1(f). (3.4) Case 1: g=σG,ω−1(e).  σG[1]e[ω]f,ω(σG,ω−1(e))=σG[1]e,ω(σG,ω−1(e))(byTheorem2.1(5,2),using(3.3))=σG,ω(e)(byTheorem2.1(2,1)). On the other hand, if e≠σG,ω(f), then   σG[ω]f[1]e,ω(σG,ω−1(e))=σG[ω]f[1]e,ω(σG[ω]f,ω−1(e))(byLemma3.4(b))=σG[ω]f,ω(e)(byTheorem2.1(2,1))=σG,ω(e)(byTheorem2.1(5,2),using(3.3)),while if e=σG,ω(f), then σG,ω−1(e)=f which is not in the domain of σG[ω]f[1]e,ω so this situation does not arise. So, in any event, σG[1]e[ω]f,ω(g)=σG[ω]f[1]e,ω(g) in this case. Case 2: g=σG,ω−1(f).   σG[1]e[ω]f,ω(σG,ω−1(f))=σG[1]e[ω]f,ω(σG[1]e,ω−1(f))(byLemma3.4(a),usingf≠σG,ω(e))=σG[1]e,ω(f)(byTheorem2.1(2,2))={σG,ω(f)(byTheorem2.1(5,1)),iff≠σG,ω−1(e),σG[1]e,ω(σG,ω−1(e)),iff=σG,ω−1(e),={σG,ω(f),iff≠σG,ω−1(e),σG,ω(e),iff=σG,ω−1(e). Now consider σG[ω]f[1]e,ω(σG,ω−1(f)). If e≠σG,ω(f), we have   σG[ω]f[1]e,ω(σG,ω−1(f))=σG[ω]f,ω(σG,ω−1(f))(byTheorem2.1(5,1)and(3.4))=σG,ω(f)(byTheorem2.1(2,2)).If e=σG,ω(f), then   σG[ω]f[1]e,ω(σG,ω−1(f))=σG[ω]f[1]e,ω(σG[ω]f,ω−1(e))(byLemma3.4(b)and(3.4))=σG[ω]f,ω(e)(byTheorem2.1(2,1))=σG,ω(e)(byTheorem2.1(5,2),usinge≠σG,ω−1(f)). Case 3: g=σG[1]e,ω−1(f).  σG[1]e[ω]f,ω(σG[1]e,ω−1(f))=σG[1]e,ω(f)(byTheorem2.1(2,2))={σG,ω(f)(byTheorem2.1(5,1)),ife≠σG,ω(f),σG[1]e,ω(σG,ω−1(e)),ife=σG,ω(f),={σG,ω(f),ife≠σG,ω(f),σG,ω(e),ife=σG,ω(f)(byTheorem2.1(2,1)). If e≠σG,ω(f), then   σG[ω]f[1]e,ω(σG[1]e,ω−1(f))=σG[ω]f[1]e,ω(σG,ω−1(f))(byLemma3.4(a),usingf≠σG,ω(e))=σG[ω]f,ω(σG,ω−1(f))(byTheorem2.1(5,1)and(3.4))=σG,ω(f)(byTheorem2.1(2,2)). If e=σG,ω(f), then   σG[ω]f[1]e,ω(σG[1]e,ω−1(f))=σG[ω]f[1]e,ω(σG,ω−1(f))(byLemma3.4(a),usingf≠σG,ω(e))=σG[ω]f[1]e,ω(σG[ω]f,ω−1(e))(byLemma3.4(b)and(3.4))=σG[ω]f,ω(e)(byTheorem2.1(2,1))=σG,ω(e)(byTheorem2.1(5,2),usinge≠σG,ω−1(f)). Case 4: g=σG[ω]f,ω−1(e). This case can be proved in a manner similar to the previous cases. But in fact this is not necessary, since we have shown the permutations σG[1]e[ω]f,ω and σG[ω]f[1]e,ω agree on every element of their common domain except one, so they must agree on this last element too.□ Lemma 3.6 If f≠σG,ω(e), then  σG[1]e[ω]f,1=σG[ω]f[1]e,1. Proof This proof is more complicated again than that of Lemma 3.5. We may suppose that neither e nor f is a triloop, since we have already established commutativity in such cases, in Lemma 3.2. So σG,μ(e)≠e and σG,μ(f)≠f, for μ∈{1,ω,ω2}. We prove that   σG[1]e[ω]f,1−1(g)=σG[ω]f[1]e,1−1(g) (3.5)for all g∈E(G)⧹{e,f}. Observe that   σG[1]e[ω]f,ω2(g)=σG[1]e,ω2(g)ifg∉{σG[1]e,ω2−1(f),σG[1]e,1(f)},byTheorem2.1(5,2)=σG,ω2(g)ifg≠σG,ω2−1(e),byTheorem2.1(5,1)=σG[ω]f,ω2(g)ifg∉{σG,ω2−1(f),σG,1(f)},byTheorem2.1(5,2)=σG[ω]f[1]e,ω2(g)ifg≠σG[ω]f,ω2−1(e),byTheorem2.1(5,1). It follows that if g∉{σG,1(f),σG,ω2−1(f),σG,ω2−1(e),σG[ω]f,ω2−1(e),σG[1]e,ω2−1(f),σG[1]e,1(f)},then   σG[1]e[ω]f,1−1(g)=σG[1]e[ω]f,ω(σG[1]e[ω]f,ω2(g))(usingσ1◦σω◦σω2=identity)=σG[1]e[ω]f,ω(σG[ω]f[1]e,ω2(g))(bythepreviousparagraph)=σG[ω]f[1]e,ω(σG[ω]f[1]e,ω2(g))(byLemma3.5)=σG[ω]f[1]e,1−1(g)(usingσ1◦σω◦σω2=identity,again). We now consider in turn how to deal with the exceptional values of g, apparently six in number. Case 1: g=σG,1(f). We must have σG,1(f)≠e, else g=e which is forbidden. We have   σG[1]e[ω]f,1−1(σG,1(f))=σG[1]e[ω]f,1−1(σG[1]e,1(f))(byTheorem2.1(5,1),usingf≠σG,ω(e))=σG[1]e,1−1(f)(byLemma3.4(d),firstcase)={σG,ω(e),iff=σG,1(e),σG,1−1(e),iff=σG,ω2−1(e),σG,1−1(f),otherwise,by Lemma 3.4(c). Now consider σG[ω]f[1]e,1−1(σG,1(f)). If f=σG,1(e), then σG[ω]f,1(e)=σG[ω]f,1(σG,1−1(f))=σG,1(f), with the second equality following from Theorem 2.1(1,2). This justifies the first step of the following:   σG[ω]f[1]e,1−1(σG,1(f))=σG[ω]f[1]e,1−1(σG[ω]f,1(e))=σG[ω]f,ω(e)(byLemma3.4(c))=σG,ω(e)(usingourhypothesis,e≠σG,ω−1(f)). If f=σG,ω2−1(e), i.e., e=σG,ω2(f), then σG[ω]f,ω2−1(e)=σG,1(f), by Lemma 3.4(f) (second case). This justifies the first step of the following:   σG[ω]f[1]e,1−1(σG,1(f))=σG[ω]f[1]e,1−1(σG[ω]f,ω2−1(e))=σG[ω]f,1−1(e)(byLemma3.4(c))=σG,1−1(e)(byLemma3.4(d),usinge≠σG,1(f)). Suppose, then, that f≠σG,1(e) and f≠σG,ω2−1(e). From e≠σG,1−1(f), we deduce that σG[ω]f,1(e)=σG,1(e), by Theorem 2.1(5,2). Also, since e≠f and σG,1 is a bijection, we have σG,1(e)≠σG,1(f). So σG[ω]f,1(e)≠σG,1(f). From e≠σG,ω2(f), and our hypothesis e≠σG,ω−1(f), we deduce from Lemma 3.4(f) that σG[ω]f,ω2−1(e)=σG,ω2−1(e). Our hypothesis e≠σG,ω−1(f) implies e≠σG,ω2(σG,1(f)), which in turn implies σG,ω2−1(e)≠σG,1(f). Combining the conclusions of the two previous sentences, we obtain σG[ω]f,ω2−1(e)≠σG,1(f). The conclusions of the previous two paragraphs, together with Lemma 3.4(c), justify the first step in the following:   σG[ω]f[1]e,1−1(σG,1(f))=σG[ω]f,1−1(σG,1(f))=σG,1−1(f)(byLemma3.4(d)). We have shown, then, that σG[1]e[ω]f,1−1 and σG[ω]f[1]e,1−1 agree on g=σG,1(f), in all circumstances. This deals with the first of our exceptional values of g. Case 2: g=σG,ω2−1(f). We must have σG,ω2−1(f)≠e, else g=e which is forbidden. First, observe that σG[1]e,1(f)∈{σG,ω2−1(e),σG,1(f)}, by Theorem 2.1(1,1), (5,1), using the hypothesis f≠σG,ω(e). Now, σG,ω2−1(e)≠σG,ω2−1(f), since e≠f. Furthermore, σG,1(f)≠σG,ω2−1(f), since if σG,1(f)=σG,ω2−1(f), then f=σG,ω2(σG,1(f))=σG,ω−1(f), which means that f is a triloop, which we excluded at the start. So, whatever its value, we have σG[1]e,1(f)≠σG,ω2−1(f). Secondly, observe that σG[1]e,ω2−1(f)=σG,ω2−1(f), by Lemma 3.4(e), using f≠σG,ω2(e) (see the start of this Case). The conclusions of these two previous paragraphs justify the first two steps in the following:   σG[1]e[ω]f,1−1(σG,ω2−1(f))=σG[1]e,1−1(σG,ω2−1(f))(byLemma3.4(d))=σG[1]e,1−1(σG[1]e,ω2−1(f))=σG[1]e,ω(f)={σG,ω(e),iff=σG,ω−1(e),σG,ω(f),otherwise,by Theorem 2.1(2,1), (5,2). Now consider σG[ω]f[1]e,1−1(σG,ω2−1(f)). If f=σG,ω−1(e), then σG,ω2−1(f)=σG,1(e). Also, e≠σG,1−1(f), since if e=σG,1−1(f) then f=σG,ω−1(σG,1−1(f))=σG,ω2(f),so that f is a triloop, which we have excluded. So σG[ω]f,1(e)=σG,1(e)=σG,ω2−1(f), with the first equality holding by Theorem 2.1(5,2). This justifies the first step in the following:   σG[ω]f[1]e,1−1(σG,ω2−1(f))=σG[ω]f[1]e,1−1(σG[ω]f,1(e))=σG[ω]f,ω(e)(byLemma3.4(c))=σG,ω(e)(byTheorem2.1(5,2),sincee≠σG,ω−1(f)byhypothesis). If f≠σG,ω−1(e), then σG,ω2−1(f)≠σG,1(e). Also, σG,ω2−1(f)≠σG,1(f), else f is a triloop, as we saw early in this Case. So σG,ω2−1(f)∉{σG,1(e),σG,1(f)}. But σG[ω]f,1(e)∈{σG,1(e),σG,1(f)}, by Theorem 2.1(1,2), (5,2). So σG,ω2−1(f)≠σG[ω]f,1(e). Since e≠σG,ω−1(f) by hypothesis, σG[ω]f,ω2−1(e)∈{σG,1(f),σG,ω2−1(e)}. Now, as we have seen, σG,ω2−1(f)≠σG,1(f), else f is a triloop; also, σG,ω2−1(f)≠σG,ω2−1(e), since e≠f. So σG,ω2−1(f)≠σG[ω]f,ω2−1(e). The conclusions of the previous two paragraphs, together with Lemma 3.4(c), justify the first step of the following:   σG[ω]f[1]e,1−1(σG,ω2−1(f))=σG[ω]f,1−1(σG,ω2−1(f))=σG,1−1(σG,ω2−1(f))(byLemma3.4(d),usingσG,ω2−1(f)≠σG,1(f))=σG,ω(f). So σG[1]e[ω]f,1−1 and σG[ω]f[1]e,1−1 agree on g=σG,ω2−1(f), in all circumstances. This deals with the second of our exceptional values of g. Case 3: g=σG,ω2−1(e). We must have σG,ω2−1(e)≠f, else g=f which is forbidden. Observe that σG[1]e,1(σG,1−1(e))=σG,ω2−1(e),by Theorem 2.1(1,1). So, if f=σG,1−1(e), then   σG[1]e[ω]f,1−1(σG,ω2−1(e))=σG[1]e[ω]f,1−1(σG[1]e,1(f))=σG[1]e,1−1(f)(byLemma3.4(d))={σG,ω(e),ifalsof=σG,1(e),σG,1−1(f),ifalsof≠σG,1(e),by Lemma 3.4(c) with f≠σG,ω2−1(e). On the other hand, if f≠σG,1−1(e), then σG[1]e,1(f)≠σG[1]e,1(σG,1−1(e)), since σG[1]e,1 is a bijection. So σG,ω2−1(e)≠σG[1]e,1(f). Therefore, we have   σG[1]e[ω]f,1−1(σG,ω2−1(e))=σG[1]e,1−1(σG,ω2−1(e))(byLemma3.4(d))=σG,1−1(e)(byLemma3.4(c)). So, in summary,   σG[1]e[ω]f,1−1(σG,ω2−1(e))={σG,ω(e),iff=σG,1−1(e)andf=σG,1(e),σG,1−1(f),iff=σG,1−1(e)andf≠σG,1(e),σG,1−1(e),iff≠σG,1−1(e). Now consider σG[ω]f[1]e,1−1(σG,ω2−1(e)). Since e≠σG,ω−1(f), by hypothesis, and e≠σG,ω2(f) (see start of this Case), Lemma 3.4(f) gives σG,ω2−1(e)=σG[ω]f,ω2−1(e). Consider, for a moment, the circumstances under which σG[ω]f,1−1(e)=e. Lemma 3.4(d) tells us that   σG[ω]f,1−1(e)={σG,1−1(f),ife=σG,1(f),σG,1−1(e),ife≠σG,1(f).If e≠σG,1(f), then σG[ω]f,1−1(e)≠e, since otherwise e=σG,1−1(e), so that e is a triloop, which we have excluded. Also, if e≠σG,1−1(f), then e cannot equal either of the two possible expressions just given for σG[ω]f,1−1(e) (using the triloop exclusion, again, for the second of these). So, again, σG[ω]f,1−1(e)≠e. On the other hand, if e=σG,1(f) and e=σG,1−1(f), then the first case above gives σG[ω]f,1−1(e)=σG,1−1(f)=e. In this situation, in applying Theorem 2.1 to G[ω]f, we cannot use case (1,1), since that would require σG[ω]f,1−1(e)≠e. Similarly, we cannot use the inverse of case (1,1) to find σG[ω]f[1]e,1−1(σG[ω]f,ω2−1(e)); instead, we must use case (5,1). If e=σG,1(f) and e=σG,1−1(f), then, we have   σG[ω]f[1]e,1−1(σG,ω2−1(e))=σG[ω]f[1]e,1−1(σG[ω]f,ω2−1(e))=σG[ω]f,1−1(σG[ω]f,ω2−1(e))(byTheorem2.1(5,1))=σG[ω]f,ω(e)=σG,ω(e)(byTheorem2.1(5,2),usingourhypothesise≠σG,ω−1(f)). Otherwise, we have   σG[ω]f[1]e,1−1(σG,ω2−1(e))=σG[ω]f[1]e,1−1(σG[ω]f,ω2−1(e))=σG[ω]f,1−1(e)(byLemma3.4(c))={σG,1−1(f),ife=σG,1(f)(andsoe≠σG,1−1(f)too,elseweareinthepreviousparagraph),σG,1−1(e),ife≠σG,1(f),by Lemma 3.4(d). So σG[1]e[ω]f,1−1 and σG[ω]f[1]e,1−1 agree on g=σG,ω2−1(e), in all circumstances. This deals with the third of our exceptional values of g. Cases 4–6: g∈{σG[ω]f,ω2−1(e),σG[1]e,ω2−1(f),σG[1]e,1(f)}. Theorem 2.1 and Lemma 3.4 tell us that   σG[ω]f,ω2−1(e)∈{σG,1(f),σG,ω2−1(e)},σG[1]e,ω2−1(f)∈{σG,ω2−1(e),σG,ω2−1(f)},σG[1]e,1(f)∈{σG,1(f),σG,ω2−1(e)}. So these are not really new cases at all; they each take us back into one of Cases 1–3. This completes our proof of (3.5), and hence of the Lemma.□ Theorem 3.7 If f≠σG,ω(e), then  G[1]e[ω]f=G[ω]f[1]e. Proof In view of Lemmas 3.5 and 3.6, we know that σG[1]e[ω]f,μ=σG[ω]f[1]e,μ for μ∈{1,ω}. But then it follows for μ=ω2 too, since σω2=σω−1◦σ1−1.□ Triality gives the following two corollaries. Corollary 3.8 If f≠σG,1(e), then  G[ω]e[ω2]f=G[ω2]f[ω]e.□ Corollary 3.9 If f≠σG,ω2(e), then  G[ω2]e[1]f=G[1]f[ω2]e.□ The results so far in this section (together with the fact of non-commutativity in general for the excluded cases for the previous three results) give us a complete description of when the μ-reductions do, or do not, commute, in general. But some interesting questions remain. Given that the excluded (generally non-commutative) cases are so specific, it is natural to ask for a characterization of those alternating dimaps for which all reductions always commute. Consider f=σG,ω(e), illustrated in Fig. 5. In this case, •[1]e and •[ω]f do not commute in general, but we can still investigate when they do. Proposition 3.10 If f=σG,ω(e), then •[1]eand •[ω]fcommute if and only if at least one of e, f is a triloop. Proof If either e or f is a triloop, then they commute by Lemma 3.2. Suppose then that neither e nor f is a triloop. If e and f form an a-face of size 2, then it is routine to show that these reductions do not commute unless the head of f meets no other edge except e, or the head of e meets no other edge except f; but that would make either e or f a 1-loop. If e and f do not form such an a-face, then the endpoints of e and f—three in number—are all distinct. The situation is then exactly as in Fig. 5 (except that the right-hand vertex might coincide with the tail of f or the head of e, but that is immaterial). It is evident from the figure that the only way the reductions can commute in this case is if the head of e has in-degree 1 (i.e., if the edges shown in green do not exist), which would make e a 1-loop.□ Theorem 3.11 Every pair of reductions on G commutes if and only if the set of triloops of G includes at least one of each pair of edges that are consecutive in any in-star, a-face or c-face. Proof Use Proposition 3.10 and triality.□ We pause now to introduce a graph derived from G which gives an alternative way of framing Theorem 3.11. The trimedial graph tri(G) of the alternating dimap G has vertex set E(G) with two vertices of tri(G) being adjacent if their corresponding edges in G are consecutive in an a-face, a c-face or an in-star of G. The trimedial graph is an alternating dimap analogue of the medial graph of a classical embedded graph. It is always undirected and 6-regular, and may have loops and/or multiple edges. Its 6-regularity implies that if it has no loops or multiple edges, then it is non-planar even if G is plane (in contrast to the usual medial graph). With this definition, we may rewrite Theorem 3.11. Corollary 3.12 Every pair of reductions on G commutes if and only if the set of triloops of G is a vertex cover of tri(G). So far, we have considered the usual kind of commutativity, where the order in which two operations are applied does not matter. We can also ask about stronger forms of commutativity. If a set of k reductions (each of the form •[μ]e, where each μ∈{1,ω,ω2} and all the e are distinct) has the property that applying them in any order always gives the same result, then we say that it is k-commutative on G. We say that G is k-reduction-commutative if every set of k reductions is k-commutative on G. It is totally reduction-commutative if it is k-reduction-commutative for every k. In this terminology, ordinary commutativity is 2-commutativity, in the sense that, if two particular reductions •[μ]e and •[ν]f commute, then the set {•[μ]e,•[ν]f} is 2-commutative. Theorem 3.11 characterizes alternating dimaps that are 2-reduction-commutative. While total reduction-commutativity implies k-reduction-commutativity for any fixed k, which in turn implies l-reduction-commutativity for any l<k, the converses do not hold. Considering how taking minors affects these properties, we see that, if G is totally reduction-commutative, then so is any minor of G. By contrast, 2-reduction-commutativity is not in general preserved by taking minors. To see this, let H be any alternating dimap with no triloops, and form G from it by inserting an ω2-loop at each vertex of each anticlockwise face and an ω-loop at each vertex of each clockwise face. Then H is a minor of G, yet Theorem 3.11 tells us that G is 2-reduction-commutative yet H is not. We now characterize alternating dimaps that are totally reduction-commutative. A 1-circuit is an alternating dimap consisting of a single directed circuit, in which every edge is a 1-loop. An ω-circuit (respectively, ω2-circuit) consists of a single vertex together with a number of ω-loops (respectively, ω2-loops) at it. A tricircuit is an alternating dimap that can be constructed from a 1-circuit, an ω-circuit and an ω2-circuit (any of which may have no edges), taking a single vertex in each, and identifying these three vertices in the natural way. This is done so as to preserve the alternating dimap property, and will entail having the ω-circuit and ω2-circuit on opposite sides of the 1-circuit. Theorem 3.13 An alternating dimap G is totally reduction-commutative if and only if each of its components is a tricircuit. Proof Suppose G is totally reduction-commutative. Then it is certainly 2-reduction-commutative, so by Theorem 3.11 the set of triloops of G includes at least one of each pair of edges that are consecutive in any in-star, a-face or c-face. Consider those edges of G which have distinct endpoints (i.e., the non-loops). Suppose two non-loop edges e and f share an endpoint v, so e,f∈I(v). Since e and f are not loops, they do not come out of v. The number of half-edges going out of v must be two greater than the number of half-edges other than e and f going into v. So there must be two half-edges going out of v that do not match (that is, are not part of the same edge as) any half-edge going into v. Let g be an edge to which one of these half-edges belongs. Without loss of generality, suppose that e,g,f occur in that order, going clockwise around v. Let the sequence of edges of I(v) which are between g and e going anticlockwise be h1,…,ha, and let the sequence of edges of I(v) which are between g and f going clockwise be i1,…,ic. Then the alternating dimap G′≔G[ω2](h1,…,ha)[ω](i1,…,ic) is left with e,g,f intact, still in this same order around v, and with no edges intervening between them any more. Then e and f are consecutive (clockwise) in the in-star at v in G′. By Theorem 3.11, this implies non-commutativity of some reductions on G′, which in turn implies that G is not totally reduction-commutative. Now suppose two non-triloop non-loops e and f are head-to-tail: say, with v = head of e = tail of f. Since e is not a 1-loop, there must be other edges at v. If all of those edges lie between e and f going clockwise, then e and f are consecutive around the clockwise face containing e, so Theorem 3.11 gives non-commutativity of some reductions, so G is not totally reduction-commutative. Similarly, if those extra edges at v all lie on the other side—between f and e going clockwise—then, again, G is not totally reduction-commutative. So there are some edges on each side. Let the edges of I(v) between f and e going anticlockwise be h1,…,hk. Then G′≔G[ω2](h1,…,hk) has e and f as consecutive edges in the anticlockwise face containing e. This gives some non-commutative reductions in G′, so G is not totally reduction-commutative. If non-triloop non-loops e and f belong to the same component of G, then let P be the shortest path, in the underlying undirected graph, from one to the other. (Note, e,f∉E(P), and P meets e and f only at the endpoints of P, by its minimality.) If all the edges of P are contracted, to give G[1]E(P), then we have e and f sharing an endpoint and we are in one of the previous two paragraphs, so G[1]E(P) is not totally reduction-commutative, so neither is G. So each component of G has at most one edge that is neither a triloop nor a loop. All the 1-loops in a component of G must lie in a single directed circuit in that component. To see this, take any 1-loop e = uv. It has a unique successor, which cannot be a loop or e would not be a 1-loop. So it must either be a 1-loop or the sole edge which is neither a triloop nor a loop. Now let us go back the other way. Consider the edges in I(u). At least one of them must be a non-loop. But if I(u) has two non-loops, then both of them are not 1-loops, and so this component has at least two edges that are neither a triloop nor a loop, which is a contradiction. So I(u) has only one non-loop, which must either be a 1-loop or the sole non-triloop non-loop. We can follow 1-loops forwards and backwards in this way until we are forced to stop. This happens when we complete a circuit, which will either be a circuit consisting entirely of 1-loops—in which case it is an entire component of G—or consisting of 1-loops except for the sole non-triloop non-loop, which we call f=wx. In the latter case, other edges may meet the head x of that special edge, but cannot meet any other vertex on the circuit. The other edges at x must all be loops, since if any is an outgoing non-loop then another must be an incoming non-loop which is then not a 1-loop either, a contradiction with the uniqueness of f. Furthermore, if any edge g at x is a proper 1-semiloop, then we can form a minor, by reduction of any ω-loops or ω2-loops that get in the way, in which f and g form a configuration that allows non-commutativity. So those other edges at x must all be ω-loops or ω2-loops. This description of the component of G, as a circuit whose edges are 1-loops with possibly one exception, and with the head of that exception holding ω-loops and ω2-loops, identifies the component as a tricircuit. So every component of G is in fact a tricircuit. Conversely, if every component of G is a tricircuit, then each component has at most one edge that is not a triloop, so any two reductions on G commute, by Theorem 3.3. Therefore, G is totally reduction-commutative. □ 4. Excluded minors for fixed genus A posy, or k-posy, is an alternating dimap with one vertex, 2k+1 edges (all loops), and two faces. Its genus is k. Up to isomorphism, there is a single 0-posy, a single 1-posy and four 2-posies. The 0-posy is just a single ultraloop. The 1-posy and the four 2-posies are shown in Fig. 6. Figure 6. View largeDownload slide The 1-posy and the four 2-posies. In each posy, the two faces are coloured grey (clockwise) and white (anticlockwise). The first three posies (top row) remain identical under reversal of all edges. The last two are reversals, and mirror images, of each other. Figure 6. View largeDownload slide The 1-posy and the four 2-posies. In each posy, the two faces are coloured grey (clockwise) and white (anticlockwise). The first three posies (top row) remain identical under reversal of all edges. The last two are reversals, and mirror images, of each other. We use posies to give an excluded minor characterization of alternating dimaps of at most a given genus. An analogous result for surface minors and orientably embedded undirected graphs is established in [14]. Theorem 4.1 A non-empty alternating dimap G has genus <kif and only if none of its minors is a disjoint union of posies of total genus k. Proof The forward implication is clear, since every such union of posies has genus k, and reductions never increase the genus. For the reverse implication, we prove by induction on ∣E(G)∣ that every non-empty G has, as a minor, a disjoint union of posies of total genus γ(G). This is true for ∣E(G)∣=1, since then G is an ultraloop, which is the 0-posy. Now suppose it is true for all alternating dimaps of <m edges, where m>1. Let G be any alternating dimap with m edges. Let e∈E(G). Now, G[1]e, G[ω]e and G[ω2]e each have m−1 edges, so by the inductive hypothesis, G[μ]e has as a minor a disjoint union of posies of total genus γ(G[μ]e), for each μ∈{1,ω,ω2}. Such a minor of G[μ]e is also a minor of G, so we see that G has such a minor, for each such μ. If γ(G[μ]e)=γ(G) for any such μ, we are done. So it remains to consider the case where γ(G[μ]e)<γ(G) (in which case γ(G[μ]e)=γ(G)−1) for each μ and each e∈E(G). The condition γ(G[1]e)<γ(G) implies that e is a proper 1-semiloop, so already we know that every edge of G is a loop that does not enclose its own face. Let e be any edge and let v be the vertex at which e is a loop. The condition γ(G[ω2]e)<γ(G) implies that e is also a proper ω-semiloop. Let F be the face on the right side of e, and let F′ be the face on the right side of the left successor e′ of e (see Fig. 7). If faces F and F′ were distinct, then ω-reduction of e would not reduce the genus and e would not be an ω-semiloop. So F=F′. Applying this same reasoning to the next edge (clockwise from e) in the in-star at v (denoted by f in Fig. 7) shows that the face F′ is, in turn, identical to the second face beyond it (denoted by F″), continuing to go clockwise around v. Continuing in this manner we find that every second ‘face’ around v is really just part of one single face. In a similar manner, the condition γ(G[ω]e)<γ(G) implies that e is also a proper ω2-semiloop, and we find that all the ‘faces’ at v which were not accounted for in the previous paragraph (being every other second face around v) are, again, just one single face (but necessarily distinct from the face F we found there). So the component of G that consists of loops at v is a posy. So G itself is just a disjoint union of all these posies, so we are done.□ Figure 7. View largeDownload slide Proof of Theorem 4.1: faces around v when e is a 1-semiloop and an ω-semiloop. Figure 7. View largeDownload slide Proof of Theorem 4.1: faces around v when e is a 1-semiloop and an ω-semiloop. 5. Tutte invariants We now extend the notion of a Tutte invariant to alternating dimaps, and investigate what invariants of this type exist. We define two types of these invariants, in each case attempting to keep the spirit of the recursive definition of the Tutte polynomial. The first seems very general, while the second is about as simple as possible. Definitions An extended Tutte invariant for alternating dimaps is a function F defined on every alternating dimap such that F is invariant under isomorphism, F(emptyalternatingdimap)=1, and there exist w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l such that, for any alternating dimap G, for any ultraloop e of G,   F(G)=wF(G[∗]e), (5.1) for any proper 1-loop e of G,   F(G)=xF(G[1]e), (5.2) for any proper ω-loop e of G,   F(G)=yF(G[ω]e), (5.3) for any proper ω2-loop e of G,   F(G)=zF(G[ω2]e), (5.4) for any proper 1-semiloop e of G,   F(G)=aF(G[1]e)+bF(G[ω]e)+cF(G[ω2]e), (5.5) for any proper ω-semiloop e of G,   F(G)=dF(G[1]e)+eF(G[ω]e)+fF(G[ω2]e), (5.6) for any proper ω2-semiloop e of G,   F(G)=gF(G[1]e)+hF(G[ω]e)+iF(G[ω2]e), (5.7) for any other edge e of G (i.e., one that is not a semiloop),   F(G)=jF(G[1]e)+kF(G[ω]e)+lF(G[ω2]e). (5.8) A simple Tutte invariant for alternating dimaps is an extended Tutte invariant for which a=b=c=d=e=f=g=h=i=j=k=l=1. In other words, proper semiloops are all treated the same as each other, and as non-semiloops, and all the coefficients in their recursive expressions are 1. We first describe all simple Tutte invariants. Theorem 5.1 The only simple Tutte invariants of alternating dimaps are the following: F(G)=0for non-empty G, with w=0; F(G)=3∣E(G)∣, with w=x=y=z=3; F(G)=(−1)∣V(G)∣, with y=z=1and x=w=−1; F(G)=(−1)af(G), with x=z=1and y=w=−1; F(G)=(−1)cf(G), with x=y=1and z=w=−1. Proof Let F be a simple Tutte invariant of alternating dimaps, with w,x,y,z as in the definition. If w=0, then it is easily shown that F(G)=0 for all non-empty G. So suppose w≠0. For any k≥1, let Uk be a disjoint union of k ultraloops. For any k≥2, let Lk,1 be the directed cycle of k vertices and k edges, which has one a-face, one c-face and k in-stars. For k≥2 and μ∈{ω,ω2}, let Lk,μ be the alternating dimap consisting of a single vertex with kμ-loops. It is clear that   F(Uk)=wk, (5.9)  F(L2,1)=xw, (5.10)  F(L2,ω)=yw, (5.11)  F(L2,ω2)=zw. (5.12) Consider the alternating dimap L2,1+eω2 obtained by adding an ω2-loop e to a vertex v of L2,1. (This is the black dimap in Fig. 1(b).) Let f (respectively, g) be the edge of L2,1 going out of (respectively, into) v. Observe that f is a 1-loop and g is not a triloop. We can calculate F(L2,1+eω2) by applying (5.2) at f (or (5.4) at e), obtaining xzw. Alternatively, we can apply (5.8) at g, obtaining (z+x+w)w. Equating the results and using w≠0, we obtain   x+z+w=xz. (5.13)Similar reasoning for the trials (L2,1+eω2)ω and (L2,1+eω2)ω2 gives   F((L2,1+eω2)ω)=xyw, (5.14)  F((L2,1+eω2)ω2)=yzw, (5.15)  x+y+w=xy, (5.16)  y+z+w=yz. (5.17) Now consider the alternating dimap obtained from L2,1 (with edges g,h) by adding, to the endpoint v of h, an ω2-loop e within the anticlockwise face and an ω-loop f within the clockwise face. Call it A. Using the ω2-loop e, or the ω-loop f, or the 1-loop g, we find that   F(A)=xyzw. (5.18)But using h, which is not a triloop, we have   F(A)=F((L2,1+eω2)ω2)+F((L2,1+eω2)ω)+F(L2,1+eω2)=yzw+xyw+xzw=(yz+xy+xz)w.Equating with (5.18), and using w≠0, we obtain   xz+xy+yz=xyz. (5.19) From (5.13), (5.16) and (5.17), we obtain   w=xz−x−z=xy−x−y=yz−y−z.The second equality here gives (x−1)z=(x−1)y, so either x=1 or y=z. Similarly, either y=1 or x=z, and either z=1 or x=y. Combining these, we have one of x=y=z, x=z=1 and y≠1, x=y=1 and z≠1, y=z=1 and x≠1.If (i) holds, then any of (5.13), (5.16) and (5.17) gives   w=x(x−2). (5.20)Also (5.19) gives 3x2=x3, whence x=3 (since x=0 would imply w=0, by (5.20)) and w=3 (by (5.20)). If (ii) holds, then (5.16) gives w=−1. Similarly, cases (iii) and (iv) give w=−1 too. Also, (5.19) implies y=−1 in case (ii), z=−1 in case (iii) and x=−1 in case (iv). We now establish the form of F for each of cases (i)–(iv) in turn. The numbering of the claims indicates the case to which each applies. Claim (i): F(G)=3∣E(G)∣. Proof of Claim (i): we use induction on ∣E(G)∣. If ∣E(G)∣=0 then the claim is true by the definition of F. Suppose ∣E(G)∣=m>1. Let e∈E(G). If e is an ultraloop, then F(G)=wF(G[∗]e)=3F(G[∗]e)=3·3m−1, by the inductive hypothesis, which equals 3m. If e is a proper μ-loop, with μ∈{1,ω,ω2}, then F(G)=xF(G[μ]e)=3·3m−1=3m. If e is not a triloop, then F(G)=F(G[1]e)+F(G[ω]e)+F(G[ω2]e)=3m−1+3m−1+3m−1=3m. Claim (ii): F(G)=(−1)af(G). Proof of Claim (ii): we use induction on ∣E(G)∣. If ∣E(G)∣=0, then the claim is true by the definition of F. Suppose ∣E(G)∣=m>1. Let e∈E(G). If e is an ultraloop, then F(G)=wF(G[∗]e)=−F(G[∗]e)=−(−1)af(G)−1=(−1)af(G). Observe that the number of anticlockwise faces in an alternating dimap goes down by 1 when an ω-loop is reduced, and it may be altered when an edge is ω2-reduced. But if e is not an ω-loop, then the number of anticlockwise faces is unchanged by 1- or ω-reduction. If e is a proper ω-loop, then F(G)=yF(G[ω]e)=−F(G[ω]e)=−(−1)af(G[ω]e)=−(−1)af(G)−1=(−1)af(G).If e is a proper μ-loop with μ∈{1,ω2}, then F(G)=F(G[μ]e)=(−1)af(G[μ]e)=(−1)af(G). If e is a proper ω-semiloop, then af(G[ω2]e)=af(G)+1, while if e is neither a triloop nor an ω-semiloop, then af(G[ω2]e)=af(G)−1.In any event, if e is not a triloop, then F(G)=F(G[1]e)+F(G[ω]e)+F(G[ω2]e)=(−1)af(G[1]e)+(−1)af(G[ω]e)+(−1)af(G[ω2]e)=(−1)af(G)+(−1)af(G)+(−1)af(G)±1=(−1)af(G). Claim (iii): F(G)=(−1)cf(G). Claim (iv): F(G)=(−1)∣V(G)∣. The proofs of Claims (iii) and (iv) are similar to that of Claim (ii), and are left as an exercise. For Claim (iv), bear in mind that ∣V(G)∣ is the number of in-stars of G.□ We now turn to extended Tutte invariants. A basic one is   F(G)=α∣E(G)∣β∣V(G)∣γaf(G)δcf(G),with α,β,γ,δ≠0. This satisfies the definition with w=αβγδ, x=αβ, y=αγ, z=αδ, a=α/β, f=α/δ, h=α/γ, b=c=d=e=g=i=0, j=αβ/3, k=αγ/3, and l=αδ/3. Extended Tutte invariants are much richer than simple Tutte invariants, since they include the Tutte polynomial for planar graphs, in a sense we now explain. The Tutte polynomial T(G;x,y) of a graph G has the following inductive definition. If E(G)=∅ then T(G;x,y)=1. Otherwise, for any e∈E(G),   T(G;x,y)={xT(G⧹e;x,y),ifeisacoloop;yT(G/e;x,y),ifeisaloop;T(G/e;x,y)+T(G⧹e;x,y),otherwise. To any orientably 2-cell-embedded (undirected) graph G, we can associate two alternating dimaps altc(G) and alta(G) as follows. For altc(G) (respectively, alta(G)), replace each edge e=uv∈E(G) by a pair of oppositely directed edges (u,v) and (v,u), forming a clockwise (respectively, anticlockwise) face of size two. The faces of G now all correspond to anticlockwise (respectively, clockwise) faces in altc(G) (respectively, alta(G)). For any alternating dimap G, define Tc(G;x,y) and Ta(G;x,y) as follows. If E(G)=∅, then Tc(G;x,y)=Ta(G;x,y)=1. Otherwise, for any e∈E(G),   Tc(G;x,y)={Tc(G[∗]e;x,y),ifeisanω2-loop(includinganultraloop);xTc(G[ω2]e;x,y),ifeisanω-semiloop;yTc(G[1]e;x,y),ifeisaproper1-semilooporanω-loop;Tc(G[1]e;x,y),+Tc(G[ω2]e;x,y),ifeisnotasemiloop.Ta(G;x,y)={Ta(G[*]e;x,y),ifeisanω-loop(includinganultraloop);xTa(G[ω]e;x,y),ifeisanω2-semiloop;yTa(G[1]e;x,y),ifeisaproper1-semilooporanω2-loop;Ta(G[1]e;x,y),+Ta(G[ω]e;x,y),ifeisnotasemiloop. Theorem 5.2 For any plane graph G,   T(G;x,y)=Tc(altc(G);x,y)=Ta(alta(G);x,y). Proof For any vertex v, write L(ω)(v) and L(ω2)(v) for an ω-loop and an ω2-loop, respectively, at v. If such a loop is added to an alternating dimap, it must be placed within a c-face or an a-face, respectively. Consider altc(G). Observe that, for any uv∈E(G),   altc(G)[1](u,v)=altc(G/uv)+L(ω2)(u′),altc(G)[ω2](u,v)=altc(G⧹uv)+L(ω2)(u′),for some u′. (Mostly u′=u, except that a little more detail is needed if (u,v) is a proper 1-semiloop, but the exact location of these extra triloops is not important.) We use these observations to prove, by induction on ∣E(G)∣, that T(G;x,y)=Tc(altc(G);x,y) for any plane graph G. It is clear from the definitions that they are identical when G is empty. Suppose then that T(G;x,y)=Tc(altc(G);x,y) when ∣E(G)∣<m, where m≥1. Let G be any plane graph on m edges, and let e=uv∈E(G). If e is a coloop, then (u,v) and (v,u) are both ω-semiloops in altc(G). (Conversely, if (u,v) and (v,u) are both proper ω-semiloops in altc(G), then uv is a coloop in G. This does not hold in general if G is not plane, however.) We have   Tc(altc(G);x,y)=xTc(altc(G)[ω2](u,v);x,y)=xTc(altc(G⧹uv)+L(ω2)(u′);x,y)=xTc(altc(G⧹uv);x,y)=xT(G⧹uv;x,y)=T(G;x,y),where the penultimate equality uses the inductive hypothesis. If e is a loop, then in altc(G) the directed versions (u,v) and (v,u) are both 1-semiloops. (This time, the converse holds even if G is not plane.) One of them may also be an ω-loop, but neither is an ω2-loop. In any case, we find that Tc(altc(G);x,y)=T(G;x,y) by a similar argument to that just used for coloops. If e is neither a coloop nor a loop, we have   Tc(altc(G);x,y)=Tc(altc(G)[1](u,v);x,y)+Tc(altc(G)[ω2](u,v);x,y)=Tc(altc(G/uv)+L(ω2)(u′);x,y)+Tc(altc(G⧹uv)+L(ω2)(u′);x,y)=Tc(altc(G/uv);x,y)+Tc(altc(G⧹uv);x,y)=T(G/uv;x,y)+T(G⧹uv;x,y)(bytheinductivehypothesis)=T(G;x,y). We conclude by induction that T(G;x,y)=Tc(altc(G);x,y) holds for all G. The proof that T(G;x,y)=Ta(alta(G);x,y) follows the same line, with appropriate adjustments.□ Having constructed alternating dimaps from embedded graphs, by replacing edges by c-faces, or by a-faces, of size 2, it is natural to ask about replacing edges by in-stars of size 2. To do this, for an embedded graph G, first construct its medial graph, med(G), then turn it into an alternating dimap by directing the edges so as to ensure the alternating property. For each component of G, there are two such ways of directing the edges in that component. There are, therefore, 2k(G) different alternating dimaps constructible from G in this way, all with med(G) as the underlying embedded graph. We refer to any one of them as alti(G). Write Ti(G;x) for any invariant of alternating dimaps that satisfies the following:   Ti(G;x)={1,ifGisempty;Ti(G[*]e;x),ifeisa1-loop(includinganultraloop);xTi(G[ω2]e;x),ifeisaproperω-semilooporanω2-loop;xTi(G[ω]e;x),ifeisaproperω2-semilooporanω-loop;Ti(G[ω]e;x)+Ti(G[ω2]e;x),ifeisnotasemiloop.This is not a full definition of a unique Ti(G;x), since we have not specified what happens if e is a proper 1-semiloop. But, since med(G) is 4-regular, alti(G) has no proper 1-semiloops. Furthermore, the only minors of it we need to form do not require 1-reduction, so these minors are each 4-regular and so have no proper 1-semiloop too. Theorem 5.3 For any plane graph G,   T(G;x,x)=Ti(alti(G);x). Proof For any alternating dimap H, write H(1) for any alternating dimap obtained from H by either adjoining an ultraloop (which becomes its own new component) or subdividing some edge (by insertion in it of a new vertex of indegree=outdegree=1, with the edge going into the new vertex being a proper 1-loop). In either case, a new 1-loop is created. Let G be a plane graph and fix any specific alti(G). If e∈E(G), write e↓ for either of the edges of alti(G) that are directed into the vertex representing e. Observe that if e∈E(G) is neither a loop nor a coloop,   {alti(G)[ω]e↓,alti(G)[ω2]e↓}={alti(G/e)(1),alti(G⧹e)(1)}.Therefore,   Ti(alti(G)[ω]e↓;x)+Ti(alti(G)[ω2]e↓;x)=Ti(alti(G/e)(1);x)+Ti(alti(G⧹e)(1);x).If e is either a coloop or a loop, then   alti(G)[ω2]e↓∈{alti(G/e)(1),alti(G⧹e)(1)},alti(G)[ω]e↓∈{alti(G/e)(1),alti(G⧹e)(1)}. We now prove the theorem by induction on ∣E(G)∣. The base case is immediate from the definition. So suppose G is a plane graph with m edges, where m≥1. If e is a coloop or a loop, then e↓ is an ω- or an ω2-semiloop in alti(G), except that it is not a 1-loop. From our (partial) definition of Ti(alti(G);x), we have   Ti(alti(G);x)∈{xTi(alti(G)[ω]e↓;x),xTi(alti(G)[ω2]e↓;x)}⊆{xTi(alti(G/e)(1);x),xTi(alti(G⧹e)(1);x)}={xTi(alti(G/e);x),xTi(alti(G⧹e);x)}={xT(G/e;x,x),xT(G⧹e;x,x)},by the inductive hypothesis. But, for such an e, the graphs G/e and G⧹e have isomorphic cycle matroids, so their Tutte polynomials are identical. Therefore,   Ti(alti(G);x)=xT(G/e;x,x)=xT(G⧹e;x,x).But these two quantities each equal T(G;x,x), for such an e, so we are done in this case. If e is neither a loop nor a coloop, then   Ti(alti(G);x)=Ti(alti(G)[ω]e↓;x)+Ti(alti(G)[ω2]e↓;x)=Ti(alti(G/e)(1);x)+Ti(alti(G⧹e)(1);x)=Ti(alti(G/e);x)+Ti(alti(G⧹e);x)=T(G/e;x,x)+T(G⧹e;x,x)(bytheinductivehypothesis)=T(G;x,x). The result follows.□ The Tutte polynomial evaluation T(G;x,x) is just the Martin polynomial of med(G) (see for example [20]). 6. Future work This work suggests many questions and problems for further research. Other combinatorial structures with triality and three minor operations As mentioned in Section 1.1, we have determined the relationship between alternating dimaps and binary functions [27]. It remains to determine the precise relationship between alternating dimaps, or indeed binary functions, and the other object types listed there. What do all the permutations generate? The mappings σH,μ, where H ranges over all minors of an alternating dimap G and μ∈{1,ω,ω2}, together generate (under composition) an inverse semigroup, which we denote by IS(G). This suggests the problem of describing IS(G) and classifying it among known types of inverse semigroup. Non-commutative minors in general Are there other types of combinatorial objects with natural minor operations that do not commute? To be convincing, such object types would need to have some of the hallmarks of a good theory of minors, such as excluded minor characterizations and Tutte invariants. Commutativity up to isomorphism So far, we have considered commutativity (or otherwise) with respect to identity: reductions commute if and only if carrying them out in each possible order gives alternating dimaps that are identical. We could also study commutativity with respect to isomorphism, and ask for a characterization of alternating dimaps G for which, for all μ1,μ2∈{1,ω,ω2} and all e,f∈E(G),   G[μ1]e[μ2]f≅G[μ2]f[μ1]e. Other excluded minor characterizations We have given a first excluded-minor result for alternating dimaps. It would be interesting to identify other natural classes of alternating dimaps that are closed under minors and find excluded minor characterizations for them. Antichains under minor inclusion Do all minor-closed classes of alternating dimaps have a finite set of excluded minors? In other words, are alternating dimaps well-quasi-ordered under the minor relation? Extended Tutte invariants We gave a full description of all simple Tutte invariants, but have not done so for extended Tutte invariants. The latter are much richer than the former, since they include the usual Tutte polynomial of an undirected abstract planar graph. But we do not yet know if they contain interesting information that takes the embedding of the dimap into account. Tutte invariants for ordered alternating dimaps One reason that Tutte invariants of alternating dimaps are more limited than Tutte invariants for graphs is the non-commutativity of the minor operations. The definitions of such invariants for alternating dimaps require the stated recursive relations to hold for reduction of any edge of the stated type, which means that the invariant will need to be unperturbable by some variations of the order of operations. These observations raise the possibility that better invariants may come from including an ordering of the edges in the object to which the invariant applies. Definitions An ordered alternating dimap is a pair (G,<), where G is an alternating dimap and < is a linear order on E(G). If (G,<) is an ordered alternating dimap and μ∈{1,ω,ω2}, then the μ-reduction (G,<)[μ] of (G,<) is the ordered alternating dimap (G[μ]e0,<′), where e0 is the first edge in E(G) under < and the order <′ on E(G)⧹{e0} is obtained by simply removing e0 from the order <. Tutte invariants and extended Tutte invariants are defined for these objects by modifying the definitions of such invariants for ordinary alternating dimaps as follows: The definitions apply to ordered alternating dimaps, rather than just to alternating dimaps. All references to G[μ]e are replaced by (G,<)[μ], for each μ. All universal quantification over edges is deleted (since there is no choice of which edge to reduce, since it is always the first edge in the ordering which must be reduced). All reference to an edge e is replaced by reference to the first edge e0 in the ordering.For example, the second condition in each of the definitions becomes: if e0 is a 1-loop, then F((G,<))=xF((G,<)[1]). When G is a general plane alternating dimap, the extended Tutte invariants Tc(G;x,y) and Ta(G;x,y) we considered earlier actually depend on the order in which the edges are considered. So they pertain to ordered alternating dimaps. But, if G has the form altc(H) (with analogous remarks applying to alta(H)), then the order in which the edges uv of H are considered does not matter, and each time a corresponding (u,v) is reduced in G, it leaves behind an ω2-loop which can be reduced at any time. (Note also that if we do not use ω-reductions, we cannot encounter those situations of non-commutativity for two edges that are consecutive in an a-face or an in-star.) For such cases, the invariants are well-defined without having to specify an order on the edges at the beginning. We ask for a characterization of (a) simple Tutte invariants, and (b) extended Tutte invariants, of ordered alternating dimaps. Funding and presentations Part of the work of this paper was done while the author was a Visiting Fellow (Combinatorics and Statistical Mechanics programme) at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January–February 2008, and on sabbatical at the Department of Mathematics and Statistics, University of Melbourne (January–June 2011). 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Wagner, Über eine Eigenschaft der ebenen Komplexe, Math. Ann.  114 ( 1937), 570– 590. Google Scholar CrossRef Search ADS   41 D. J. A. Welsh, Complexity: Knots, Colourings and Counting, London Math. Soc. Lecture Note Series 186 , Cambridge University Press, Cambridge, 1993. Google Scholar CrossRef Search ADS   © 2017. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Quarterly Journal of Mathematics Oxford University Press

Minors for alternating dimaps

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Abstract

Abstract We develop a theory of minors for alternating dimaps—orientably embedded digraphs where, at each vertex, the incident edges (taken in the order given by the embedding) are directed alternately into, and out of, the vertex. We show that they are related by the triality relation of Tutte. They do not commute in general, though do in many circumstances, and we characterize the situations where they do. We give a characterization of alternating dimaps of at most a given genus, using a finite set of excluded minors. We also use the minor operations to define simple Tutte invariants for alternating dimaps and characterize them. We establish a connection with the Tutte polynomial, and pose the problem of characterizing universal Tutte-like invariants for alternating dimaps based on these minor operations. 1. Introduction The minor relation is one of the most important order relations on graphs. A graph H is a minor of a graph G if it can be obtained from G by some sequence of deletions and contractions of edges. Many important classes of graphs can be characterized by the exclusion of some finite set of minors. These include forests, series-parallel graphs [15, 19], outerplanar graphs [12], planar graphs [30, 40]—and, in fact, any minor-closed class of graphs, by Robertson and Seymour’s proof of Wagner’s conjecture [35]. Minors also play a central role in enumerative graph theory: the Tutte–Whitney polynomials, which contain information on a great variety of counting problems on graphs or matroids, satisfy recurrence relations using deletion and contraction (see, for example [9, 21, 24, 41]). The theory of minors derives much of its richness and beauty from the fact that the deletion and contraction operations are dual (in the sense of planar graph duality or, more generally, matroid duality [34]) and commute. In this paper, we introduce and study a minor relation on alternating dimaps. An alternating dimap is a directed graph without isolated vertices, 2-cell-embedded in a disjoint union of orientable 2-manifolds, where each vertex has even degree and, for each vertex v, the edges incident with v are directed alternately into, and out of, v (when considered in the order in which they appear around v in the embedding). An alternating dimap may have loops and/or multiple edges, but cannot have a bridge. We allow the empty alternating dimap with no vertices, edges or faces. For alternating dimaps, we have three minor operations, instead of two. We show in Section 2 that they are related by a triality relation of Tutte [37], in a manner analogous to the duality between deletion and contraction. The form of the relationship is the same as that found by the author for some other combinatorial objects (binary functions) on which minors and triality can be defined [26]. One property of ordinary minor operations (and also of the minor operations in [26]) is that they commute. We show in Section 3 that minor operations on alternating dimaps do not commute in general, although they do in most circumstances, and we determine exactly when they do. As seen in the first paragraph, two of the main themes of the classical theory of minors are excluded minor characterizations and Tutte invariants. The remainder of this paper takes these themes up for alternating dimaps and their minor operations. In Section 4, we give an excluded minor characterization of alternating dimaps of at most a given genus, using a finite set of excluded minors in every case. In Section 5, we define simple Tutte invariants for alternating dimaps, and show that there are only a few of them and that they do not contain much information, in contrast to the situation for graphs, matroids and binary functions. We then define extended Tutte invariants and raise the question of how many and varied they might be. We show that they are much richer than the simple Tutte invariants, as they include, in a sense, the Tutte polynomial of a planar graph. Sections 3–5 may be read independently of each other. 1.1. History and related work Plane alternating dimaps were studied by Tutte [37, 38]. He showed that they come in triples, with the three members of a triple being derivable from a single larger structure, a bicubic map (see below). The relationship among the members of such a triple is called triality [37] or trinity [38], and each is trial or trine to the others. This relationship extends ordinary duality. Tutte’s original motivation was to determine when equilateral triangles can be tessellated by smaller equilateral triangles of different sizes or orientations. He also proved his Tree Trinity Theorem, on spanning arborescences of such maps. In the second paper [38], he noted the possibility of extending this theory to other surfaces. Berman [2] showed explicitly how to construct the trial of an alternating dimap without reference to the bicubic map from which three trial maps are derived, and gave alternative proofs of some of Tutte’s results. Tutte reviewed some aspects of his theory in [39]. Tutte showed that a ‘triangulated triangle’ gives rise to a plane bipartite cubic graph (a plane bicubic map), which in turn has, as its dual, an Eulerian plane triangulation. The triple of trial alternating dimaps is derived from this bicubic map. Although bicubic maps are plane in Tutte’s work, they may more generally be taken to be embedded in some orientable surface, so each has a genus. It is interesting to note that this stream of research, first seen in Tutte’s 1948 paper [37], can be traced back to the same source that eventually gave rise to Tutte’s work on minor operations and his eponymous polynomial. Historically, the source of both streams was the famous paper on ‘squaring the square’ [7]. The 1948 paper extended the theory to ‘triangulating the triangle’ (where all triangles are equilateral) and introduced triality, among other things. However, this stream has not previously seen the development of minor operations or Tutte-like invariants for alternating dimaps. In one of their later and lesser-known papers, Brooks et al. introduced reductions for plane bicubic maps that correspond to ours [8, Section 11], although they are not translated into alternating dimap form, are not treated as minor operations, and are used for a different purpose. Jaeger [29] used related reductions on plane bicubic maps. A separate stream of research concerns latin bitrades, which are pairs of partial latin squares of the same shape and with the same symbol set in each row and column. These may also be given a genus. Cavenagh and Lisoněk [10] established a correspondence between spherical latin bitrades and 3-connected planar Eulerian triangulations (dually, 3-connected plane bicubic maps), while the relationship between spherical latin bitrades and triangulated triangles is described in [16, 17] (see also [11, 18]). Batagelj [1] introduced two operations on plane Eulerian triangulations by means of which larger such maps can be generated from smaller ones. These operations have been used and extended in several papers (via the aforementioned correspondences) to generate latin bitrades [17, 28]. The inverse of one of these operations, translated to alternating dimaps, corresponds to (technically, restricted versions of) our minor operations. (I thank Tony Grubman and Ian Wanless for pointing out this link.) Although duality, with associated minors, appears in many forms for many different kinds of objects, there are far fewer settings with natural minor operations related by triality. The main ones known to the author are alternating dimaps, binary functions [26] (see also [22, 23, 24, 25]), multimatroids (including isotropic systems) [5, 6] and the related transition matroids [36, pp. 8, 10]. Some alternating dimaps certainly cannot be represented by objects of these other types, since alternating dimap minor operations may not commute, unlike those in the other settings. In [27, Section 4], we determine those alternating dimaps that can be represented faithfully by binary functions. 1.2. Definitions and notation If G is an alternating dimap, then kG is the disjoint union of k copies of G. Each of these copies is regarded as being embedded in a different surface, with all these k surfaces being disjoint from each other. An edge e from u to v is sometimes written e(u,v). Let G be an alternating dimap, viewed topologically as embedded in an orientable surface. Let C⊆E(G) be a circuit of G (which need not have all its edges directed the same way), and let S be the connected surface in which the component of G containing C is embedded. Then the sides of C are the components of S−C. A side is planar if it is homeomorphic to the open unit disc. The genus γ(G) of an alternating dimap G is given by   ∣V(G)∣−∣E(G)∣+∣F(G)∣=2(k(G)−γ(G)),where F(G) is the set of faces of G and k(G) is the number of components of G. The edges around a face all go in the same direction, and we say the face is clockwise or anticlockwise according to the direction of the edges around it. We identify a clockwise (respectively, anticlockwise) face with its cyclic sequence of edges, and call it a c-face (respectively, a-face) for short. Observe that the (edge sets of the) c-faces partition E(G), as do the a-faces. So every edge e belongs to one c-face, denoted by C(e)=CG(e), and one a-face, denoted by A(e)=AG(e). If two faces share a common edge, then one of the faces is clockwise and the other is anticlockwise. The left successor (respectively, right successor) of an edge e is the next edge along from e, going around AG(e) (respectively, CG(e)) in the direction given by e. (This direction is anticlockwise for the left successor, and clockwise for the right successor.) Often, c-faces and a-faces are simple cycles, but this is not always the case. If v is a cutvertex of G, then one face incident with v consists of two or more edge-disjoint cycles. The numbers of clockwise and anticlockwise faces of G are denoted by cf(G) and af(G), respectively. An in-star is the set of all edges directed into some vertex. So in-stars are in one-to-one correspondence with vertices. The in-star of edges directed into vertex v is denoted by I(v)=IG(v). Observe that the in-stars partition E(G), so every edge e also belongs to one in-star, denoted by I(e)=IG(e) (overloading notation slightly). If an alternating dimap is disconnected, then we treat its components as being embedded in separate, disjoint surfaces. Alternating dimaps extend ordinary embedded graphs (in orientable surfaces), in that replacing each edge of an embedded graph by a pair of directed edges, forming a clockwise face of size 2, gives an alternating dimap [38]. An alternating dimap G defines three permutations σG,1,σG,ω,σG,ω2:E(G)→E(G) (abbreviated σ1,σω,σω2 where G is clear from the context), as follows. For each e∈E(G), its image under σG,1, σG,ω and σG,ω2 is the next edge in clockwise order around IG(e), AG(e) and CG(e), respectively. So the left successor of e is σG,ω−1(e), while the right successor of e is σG,ω2(e). Note that, in going around an in-star in clockwise order, we skip outgoing edges at the vertex as these do not belong to the in-star. While ω can often be treated just as a symbol, at times we do algebra with it, using the value ω=exp(2πi/3). Any two of our three permutations determine the other. This follows from the relation σ1σωσω2=idE(G), the identity permutation on E(G) (with permutations applied from right to left). Let E be any finite set and let SE be the set of all triples (σ0,σ1,σ2) of permutations, each acting on E, such that σ0σ1σ2=idE. This is called a 3-constellation or a hypermap [31]. When one of the permutations is an involution, we may take its cycles to correspond to undirected edges (using the aforementioned representation of embedded graphs by alternating dimaps), and we have a standard combinatorial representation of an orientably embedded graph (see, for example [4, Section 2.2]). In the general case, we have an equivalence with alternating dimaps on E (i.e., whose edges are labelled by E) which seems to be well known (see for example [13]) although I have not seen it stated explicitly. Proposition 1.1 The map {alternatingdimapsonE}→SEgiven by G↦(σG,1,σG,ω,σG,ω2)is a bijection.□ If G is an alternating dimap, then the trial Gω of G is defined as follows. Its vertices represent the c-faces of G. We denote the vertex of Gω representing c-face C by vC. (Think of vC being placed inside C in the embedding.) Edges of Gω are constructed as follows. Suppose two c-faces C1 and C2 of G share a vertex v, and that there is an a-face A containing edges e and f going into and out of v, respectively, with e and f also belonging to C1 and C2, respectively. (See Fig. 1(a). We do not require e and f to be distinct, or C1 and C2 to be distinct.) Then we put an edge eω from vC2 to vC1 in Gω. Figure 1. View largeDownload slide Construction of trial dimap Gω from G: (a) clockwise faces → vertices and e↦eω; (b) example showing G (solid edges), Gω (dashed edges) and Gω2 (dotted edges). Note that Gω3=G. Figure 1. View largeDownload slide Construction of trial dimap Gω from G: (a) clockwise faces → vertices and e↦eω; (b) example showing G (solid edges), Gω (dashed edges) and Gω2 (dotted edges). Note that Gω3=G. These edges of Gω are ordered around C1 according to the order of the edges e around C1. Similarly, they are ordered around C2 according to the order of the edges f around C2. (Think of eω as being drawn by a curve from vC2, inside C2, to its destination vC1 inside C1, in such a way that it crosses f in its ‘first half’ (i.e., closer to its start than its end) and crosses e in its ‘last half’.) An example of the construction, for an alternating dimap on three edges, is given in Fig. 1(b). It is routine to show that the map •ω:E(G)→E(Gω), e↦eω is a bijection, and that the c-faces, a-faces and in-stars of Gω are the a-faces, in-stars and c-faces, respectively, of G. We can also express this relationship in the language of the permutation triples. The first permutation σGω,1 in the permutation triple for Gω represents (by its cycles) the in-stars of Gω. These correspond to c-faces of G, which are represented by (the cycles of) σG,ω2. Proposition 1.2   σG,1(e)ω=σGω,ω(eω) (1.1)  σG,ω(e)ω=σGω,ω2(eω) (1.2)  σG,ω2(e)ω=σGω,1(eω). (1.3)□ If G is represented by (σG,1,σG,ω,σG,ω2), then its trial Gω is represented by (σG,ω2,σG,1,σG,ω). It is clear that Proposition 1.2 still holds if σ is replaced by σ−1 throughout. The trial operation on a component of G is independent of the other components. We write Gω2 for (Gω)ω. From the way triality changes c-faces to in-stars to a-faces to c-faces, we find that Gω3=(Gω2)ω=G1=G. 1.3. Loops and semiloops A (standard) loop is just an edge of G which is a loop in the undirected version of G. If it is a separating circuit of the embedding, then it divides its component of the embedding surface into two sides, its clockwise side and its anticlockwise side. If it is non-separating, then it has just one side, which we take to be both its clockwise and anticlockwise side. A 1-loop is an edge whose head has degree 2. This does not need to be a standard loop, since its two vertices need not coincide. It is an edge whose left and right successors are identical, and, in such cases, we can refer unambiguously to its successor. An ω-loop is an edge forming a single-edge a-face. An ω2-loop is an edge forming a single-edge c-face. Unlike 1-loops, ω-loops and ω2-loops are standard loops. However, not every standard loop is of this type, as we will see when considering 1-semiloops shortly. For any alternating dimap G, an edge e is a 1-loop in G if and only if eω is an ω-loop in Gω, which in turn holds if and only if eω2 is an ω2-loop in Gω2. A triloop is an edge which is a μ-loop for some μ∈{1,ω,ω2}. An ultraloop is a triloop which (together with its vertex) constitutes a component of the graph. It has two faces, a c-face and an a-face, and its vertex has degree 2, so it is simultaneously a 1-loop, an ω-loop and an ω2-loop. In fact, if an edge is a μ-loop for μ equal to any two of {1,ω,ω2}, then it is an ultraloop. A 1-loop is only a standard loop if it is an ultraloop. A μ-loop is a proper μ-loop, and a proper triloop, if it is not also an ultraloop. In such a case, it is not a ν-loop for any ν∈{1,ω,ω2}⧹{μ}. A 1-semiloop is just a standard loop. An ω-semiloop is an ω2-loop or an edge e such that deleting both e and its right successor σω2(e) either increases the number of components of G or decreases its genus. (After deletion, we no longer have an alternating dimap in general; we are really referring to the underlying undirected embedded graph here, rather than G itself.) This latter condition may be written: k(G⧹{e,σω2(e)})−γ(G⧹{e,σω2(e)})>k(G)−γ(G). Similarly, an ω2-semiloop is an ω-loop or an edge e such that deleting both e and its left successor σω−1(e) either increases the number of components of G or decreases its genus. For each μ∈{1,ω,ω2}, a proper μ-semiloop is a μ-semiloop that is not a triloop. A proper 1-semiloop e either gives a non-contractible closed curve in the embedding, or each of its two sides contains an edge other than e from the same component as e. An edge is a μ-semiloop in G if and only if it is a μω-semiloop in Gω. If μ1≠μ2, an edge is both a μ1-semiloop and a μ2-semiloop if and only if it is a (μ1μ2)−1-loop. The effect of the trial construction on each type of loop can be seen in Fig. 1(b), where the initial dimap G contains an ω2-loop, a 1-loop and an ω-semiloop (each proper). 2. Minors Let G be an alternating dimap and e=e(u,v)∈E(G). If e is not a loop, then the dimap G[1]e is formed by deleting the edge e and identifying its endpoints, while preserving the order of the edges and faces around vertices. If e is an ω-loop or an ω2-loop, then G[1]e is formed just by deleting e. If e is a 1-semiloop, then G[1]e is formed as follows. Let the edges incident with v, in cyclic clockwise order around v starting with e directed into v, be e,a1,b1,…,ak,bk,e,c1,d1,…,cl,dl. Here, each ai and each di is directed out of v, while each bi and ci is directed into v. We replace v by two new vertices, v1 and v2, and reconnect the edges ai,bi,ci,di as follows. The tail of each ai and the head of each bi becomes v1 instead of v, while the head of each ci and the tail of each di becomes v2 instead of v. The edge e is deleted. The cyclic orderings of edges around v1 and v2 are those induced by the ordering around v. Everything else is unchanged. Observe that if e is a separating 1-semiloop then, in effect, the shrinking of the loop severs the subdimap in its clockwise side from that in its anticlockwise side. Each of these subdimaps is now on a separate surface, so the number of components has increased by 1. If e is not separating then, in effect, shrinking the loop cuts one of the handles of the surface component in which it is embedded, so reducing the genus by 1. This operation is called 1-reduction or contraction. See Fig. 2(a,b). It just adapts the usual contraction operation for surface minors to the alternating dimap context. Let f=f(v,w1) be the right successor of e, and let g=g(v,w2) be the left successor of e. (It is possible that f=g, in which case w1=w2. This occurs when v has indegree=outdegree=1, i.e., when e is a 1-loop.) Figure 2. View largeDownload slide Minor operations: G and reductions (black, solid edges, filled vertices), with their trials, which equal Gω and reductions (dashed edges, open vertices). (a) G and Gω, (b) G[1]e and (G[1]e)ω = Gω[ω2]eω, (c) G[ω]e and (G[ω]e)ω = Gω[1]eω, (d) G[ω2]e and (G[ω2]e)ω = Gω[ω]eω. Figure 2. View largeDownload slide Minor operations: G and reductions (black, solid edges, filled vertices), with their trials, which equal Gω and reductions (dashed edges, open vertices). (a) G and Gω, (b) G[1]e and (G[1]e)ω = Gω[ω2]eω, (c) G[ω]e and (G[ω]e)ω = Gω[1]eω, (d) G[ω2]e and (G[ω2]e)ω = Gω[ω]eω. The graph G[ω]e is formed by deleting e and, if e≠g, changing g so that it now joins u to w2. The revised edge g replaces e and g in A(e), and it replaces g in I(w2). If degv=2, then v and I(v) no longer exist in G[ω]e. If degv≠2, then the in-star at v in G[ω]e is I(v)⧹{e}. The c-face containing g in G[ω]e is (C(e)⧹{e})∪C(g). This operation is called ω-reduction. See Fig. 2(a,c). The graph G[ω2]e is formed by deleting e and, if e≠f, changing f so that it now joins u to w1. The revised edge f replaces e and f in C(e), and it replaces f in I(w1). If degv=2, then v and I(v) no longer exist in G[ω2]e. If degv≠2, then the in-star at v in G[ω2]e is I(v)⧹{e}. The a-face containing f in G[ω2]e is (A(e)⧹{e}∪A(f)). This operation is called ω2-reduction. See Fig. 2(a,d). For all μ∈{1,ω,ω2}, μ-reduction of a proper μ−1-semiloop either increases the number of connected components or decreases the genus. Each of these three operations is a reduction or a minor operation. The operations of ω-reduction and ω2-reduction are special cases of lifting (see, for example [32]), used in the immersion relation on graphs [33], here restricted to cases where the two incident edges are consecutive in a face. An alternating dimap obtained from G by a sequence of minor operations is a minor of G. If e is a triloop, then G[1]e=G[ω]e=G[ω2]e. We sometimes write G[∗]e for the common result of the three reductions in this case, in order to avoid being unnecessarily specific. If X=(x1,…,xk) is a sequence of edges, then we write G[μ]X as a shorthand for G[μ]x1[μ]x2⋯[μ]xk. It is straightforward to translate the above constructions for the minor operations into the language of permutation triples. Theorem 2.1 If G is an alternating dimap with permutation triple (σ1,σω,σω2), then the permutation triples for the three minors of G are as given in the following table. G[1]e  G[ω]e  G[ω2]e  σG[1]e,1(σ1−1(e))=σω2−1(e)σG[1]e,ω(σω−1(e))=σω(e)σG[1]e,ω2(σω2−1(e))=σω2(e)σG[1]e,1(σω(e))=σ1(e)Otherwise:σG[1]e,μ(f)=σμ(f)  σG[ω]e,1(σ1−1(e))=σ1(e)σG[ω]e,ω(σω−1(e))=σω(e)σG[ω]e,ω2(σω2−1(e))=σω−1(e)σG[ω]e,ω2(σ1(e))=σω2(e)σG[ω]e,μ(f)=σμ(f)  σG[ω2]e,1(σ1−1(e))=σ1(e)σG[ω2]e,ω(σω−1(e))=σ1−1(e)σG[ω2]e,ω2(σω2−1(e))=σω2(e)σG[ω2]e,ω(σω2(e))=σω(e)σG[ω2]e,μ(f)=σμ(f)  G[1]e  G[ω]e  G[ω2]e  σG[1]e,1(σ1−1(e))=σω2−1(e)σG[1]e,ω(σω−1(e))=σω(e)σG[1]e,ω2(σω2−1(e))=σω2(e)σG[1]e,1(σω(e))=σ1(e)Otherwise:σG[1]e,μ(f)=σμ(f)  σG[ω]e,1(σ1−1(e))=σ1(e)σG[ω]e,ω(σω−1(e))=σω(e)σG[ω]e,ω2(σω2−1(e))=σω−1(e)σG[ω]e,ω2(σ1(e))=σω2(e)σG[ω]e,μ(f)=σμ(f)  σG[ω2]e,1(σ1−1(e))=σ1(e)σG[ω2]e,ω(σω−1(e))=σ1−1(e)σG[ω2]e,ω2(σω2−1(e))=σω2(e)σG[ω2]e,ω(σω2(e))=σω(e)σG[ω2]e,μ(f)=σμ(f) □ For any reduction and any edge, if none of the first four equations in the appropriate column of the table apply, then the last one in the column is used (where, always, f∈E⧹{e}). Each equation in the table can only be used when the argument and image of the permutation both belong to the edge set of the minor, that is to E⧹{e}. So, if some σμ(e) or σμ−1(e) actually equals e, then that equation does not apply in that case. This can only happen when e is a triloop. We will refer to a specific equation in the table by ‘Theorem 2.1 (r,c)’, where r and c index the row and column in which the equation appears. For example, Theorem 2.1(2,1) is σG[1]e,ω(σω−1(e))=σω(e), and Theorem 2.1(5,2) is σG[ω]e,μ(f)=σμ(f) which holds when the pair μ,f is not covered by any of the previous entries in that column. We are now in a position to establish the relationship between minors and triality. Theorem 2.2 If e∈E(G)and μ,ν∈{1,ω,ω2}then  Gμ[ν]eμ=(G[μν]e)μ. Theorem 2.2 extends the classical relationship between duality and minors:   G*⧹e=(G/e)*,G*/e=(G⧹e)*.The analogy between the two settings is shown in Figs. 3 and 4. The former shows the classical relationship, while the latter illustrates Theorem 2.2. Proof of Theorem 2.2 The case μ=1 is trivial. We begin with μ=ω, and first prove that   Gω[1]eω=(G[ω]e)ω. (2.1) It is straightforward when e is an ultraloop. So suppose that e is not an ultraloop.   σGω[1]eω,1(σGω,1−1(eω))=σGω,ω2−1(eω)(byTheorem2.1(1,1))=σG,ω−1(e)ω(by(1.2))=σG[ω]e,ω2(σG,ω2−1(e))ω(byTheorem2.1(3,2))=σ(G[ω]e)ω,1(σG,ω2−1(e)ω)(by(1.3))=σ(G[ω]e)ω,1(σGω,1−1(eω))(by(1.3)).Similarly, we have   σGω[1]eω,ω(σGω,ω−1(eω))=σGω,ω(eω)=σG,1(e)ω=σG[ω]e,1(σG,1−1(e))ω=σ(G[ω]e)ω,ω(σG,1−1(e)ω)=σ(G[ω]e)ω,ω(σGω,ω−1(eω)),σGω[1]eω,ω2(σGω,ω2−1(eω))=σGω,ω2(eω)=σG,ω(e)ω=σG[ω]e,ω(σG,ω−1(e))ω=σ(G[ω]e)ω,ω2(σG,ω−1(e)ω)=σ(G[ω]e)ω,ω2(σGω,ω2−1(eω)),σGω[1]eω,1(σGω,ω(eω))=σGω,1(eω)=σG,ω2(e)ω=σG[ω]e,ω2(σG,1(e))ω=σ(G[ω]e)ω,1(σG,1(e)ω)=σ(G[ω]e)ω,1(σGω,ω(eω)). It remains to consider the cases where   (μ,fω)∈{(1,σGω,1−1(eω)),(ω,σGω,ω−1(eω)),(ω2,σGω,ω2−1(eω)),(1,σGω,ω(eω))}. (2.2)Equivalently,   (μω2,f)∈{(1,σG,1−1(e)),(ω,σG,ω−1(e)),(ω2,σG,ω2−1(e)),(ω2,σG,1(e))}. (2.3)In these cases, we have   σGω[1]eω,μ(fω)=σGω,μ(fω)(byTheorem2.1(5,1),using(2.2))=σG,μω2(f)ω(by(1.1)–(1.3))=σG[ω]e,μω2(f)ω(byTheorem2.1(5,2),using(2.3))=σ(G[ω]e)ω,μ(fω)(by(1.1)–(1.3)). We have now shown that, for all μ∈{1,ω,ω2} and f∈E(G),   σGω[1]eω,μ(fω)=σ(G[ω]e)ω,μ(fω).It follows that Gω[1]eω and (G[ω]e)ω have the same permutation triple and so are equal. Similar arguments show that   Gω[ω]eω=(G[ω2]e)ω,Gω[ω2]eω=(G[1]e)ω.From these, it follows that   Gω2[1]eω2=(G[ω2]e)ω2,Gω2[ω]eω2=(G[1]e)ω2,Gω2[ω2]eω2=(G[ω]e)ω2.□ Figure 3. View largeDownload slide The relationship between the ordinary duality and minor operations. Figure 3. View largeDownload slide The relationship between the ordinary duality and minor operations. Figure 4. View largeDownload slide The relationship between triality and the three minor operations. The diagram wraps around at its left and right sides. Arrows downwards from G,Gω,Gω2 are minor operations; the other arrows (all rightward) are triality. Figure 4. View largeDownload slide The relationship between triality and the three minor operations. The diagram wraps around at its left and right sides. Arrows downwards from G,Gω,Gω2 are minor operations; the other arrows (all rightward) are triality. 3. Non-commutativity Deletion and contraction are well known to commute, in the sense that, for any graph G and any distinct e,f∈E(G), we have   G⧹e⧹f=G⧹f⧹e,G/e/f=G/f/e,G⧹e/f=G/f⧹e.The variants of these operations for embedded graphs, where deletion/contraction of an edge is accompanied by appropriate modifications to the embedding, also commute (see for example [3, Section 2.3]). Perhaps surprisingly, the reductions we have introduced for alternating dimaps do not always commute. Figure 5 illustrates the fact that, in general, if f=σG,ω(e), then G[1]e[ω]f≠G[ω]f[1]e. By triality, it follows that if f=σG,ω2(e), then in general G[ω2]e[1]f≠G[1]f[ω2]e, and if f=σG,1(e), then in general G[ω]e[ω2]f≠G[ω2]f[ω]e. Figure 5. View largeDownload slide Non-commutativity of minor operations. Figure 5. View largeDownload slide Non-commutativity of minor operations. In this section, we prove that these three situations are the only ones in which the reductions do not commute. We first show that two reductions of the same type always commute. Theorem 3.1 For all μ∈{1,ω,ω2},   G[μ]e[μ]f=G[μ]f[μ]e. Proof We show that   G[1]e[1]f=G[1]f[1]e, (3.1)which takes up most of the proof, and then use triality to complete it. To show (3.1), we will show that, for all μ∈{1,ω,ω2} and all g∈E(G)⧹{e,f},   σG[1]e[1]f,μ(g)=σG[1]f[1]e,μ(g). (3.2) We first do this for μ∈{ω,ω2}, which we now assume. Most situations are covered by the following reasoning:   σG[1]e[1]f,μ(g)=σG[1]e,μ(g)ifg≠σG[1]e,μ−1(f)=σG,μ(g)ifg≠σG,μ−1(e)=σG[1]f,μ(g)ifg≠σG,μ−1(f)=σG[1]f[1]e,μ(g)ifg≠σG[1]f,μ−1(e),by four applications of Theorem 2.1(5,1), since the conditions on g ensure that cases (2,1) and (3,1) (according as μ=ω or μ=ω2) of that theorem do not apply. We now deal with situations where the above conditions on g are not met. We have, apparently, four exceptional values of g. We consider each in turn. First, suppose g=σG,μ−1(e). In this case, we must assume that f≠σG,μ−1(e), else g=f and g∈domσG[1]e[1]f,μ. Consider σG[1]e[1]f,μ(g). If f=σG,μ(e), then σG[1]e,μ−1(f)=σG,μ−1(e), by Theorem 2.1(2,1), (3,1). This justifies the first step in the following:   σG[1]e[1]f,μ(σG,μ−1(e))=σG[1]e[1]f,μ(σG[1]e,μ−1(f))=σG[1]e,μ(f)(byTheorem2.1(2,1)or(3,1))=σG,μ(f)(byTheorem2.1(5,1),sincef≠σG,μ−1(e)). On the other hand, if f≠σG,μ(e), then σG[1]e,μ−1(f)=σG,μ−1(f), by Theorem 2.1(5,1). This in turn does not equal σG,μ−1(e), since e≠f and σG,μ−1 is a bijection. So   σG[1]e[1]f,μ(σG,μ−1(e))=σG[1]e,μ(σG,μ−1(e))=σG,μ(e),by Theorem 2.1(5,1), then (2,1) or (3,1). Now consider σG[1]f[1]e,μ(g). Observe that σG[1]f,μ−1(e)=σG,μ−1(e), by Theorem 2.1(5,1), since f≠σG,μ−1(e). Therefore,   σG[1]f[1]e,μ(σG,μ−1(e))=σG[1]f[1]e,μ(σG[1]f,μ−1(e))=σG[1]f,μ(e)(byTheorem2.1(2,1)or(3,1))={σG,μ(f),ife=σG,μ−1(f),σG,μ(e),otherwise,by Theorem 2.1(2,1) or (3,1), and (5,1). So σG[1]e[1]f,μ(g)=σG[1]f[1]e,μ(g) when g=σG,μ−1(e). Secondly, suppose g=σG,μ−1(f). This can be treated the same as the first case, except that e and f are swapped throughout. Thirdly and fourthly, the remaining two exceptional values of g, namely σG[1]e,μ−1(f) and σG[1]f,μ−1(e), are really nothing new, for application of Theorem 2.1 gives   σG[1]e,μ−1(f)={σG,μ−1(e),iff=σG,μ(e),σG,μ−1(f),otherwise;σG[1]f,μ−1(e)={σG,μ−1(f),ife=σG,μ(f),σG,μ−1(e),otherwise.Thus, in any event, each of these two values of g actually falls into one of the first two cases. This completes the treatment of the exceptional values of g (apparently four in number, but really just two). We have now proved (3.2) for μ∈{ω,ω2}. But it then follows immediately for μ=1 too, since σH,1=σH,ω2−1◦σH,ω−1 for any H. So (3.2) holds for all μ and all g, which establishes (3.1). Now that we know contractions commute, we can use triality to show that any two reductions of the same type commute. For any μ∈{1,ω,ω2},   G[μ]e[μ]f=(Gμ[1]eμ[1]fμ)μ−1=(Gμ[1]fμ[1]eμ)μ−1=G[μ]f[μ]e,using Theorem 2.2 twice.□ We next show that the reductions always commute if one of the edges involved is a triloop. Lemma 3.2 If f is a triloop, then for any ν∈{1,ω,ω2},   G[1]e[ν]f=G[ν]f[1]e. Proof If f is a triloop, then any ν-reduction of f just amounts to contraction of f. So   G[1]e[ν]f=G[1]e[1]f=G[1]f[1]e=G[ν]f[1]e,where the middle equality follows from Theorem 3.1.□ Theorem 3.3 If f is a triloop and μ,ν∈{1,ω,ω2}, then  G[μ]e[ν]f=G[ν]f[μ]e. Proof   G[μ]e[ν]f=(Gμ[1]eμ[νμ−1]fμ)μ−1(byTheorem2.2)=(Gμ[νμ−1]fμ[1]eμ)μ−1(byLemma3.2)=G[ν]f[μ]e(byTheorem2.2).□ Most of the remainder of this section is devoted to showing that the two different reductions commute in the remaining situations not covered in Fig. 5 and its two trials. We will need some lemmas. The first uses Theorem 2.1 to describe the inverses of the permutations representing the three minors. Lemma 3.4 (a)   σG[1]e,ω−1(h)={σG,ω−1(e),ifh=σG,ω(e);σG,ω−1(h),otherwise.(b)   σG[ω]f,ω−1(h)={σG,ω−1(f),ifh=σG,ω(f);σG,ω−1(h),otherwise.(c)   σG[1]e,1−1(h)={σG,ω(e),ifh=σG,1(e);σG,1−1(e),ifh=σG,ω2−1(e);σG,1−1(h),otherwise.(d)   σG[ω]f,1−1(h)={σG,1−1(f),ifh=σG,1(f);σG,1−1(h),otherwise.(e)   σG[1]e,ω2−1(h)={σG,ω2−1(e),ifh=σG,ω2(e);σG,ω2−1(h),otherwise.(f)   σG[ω]f,ω2−1(h)={σG,ω2−1(f),ifh=σG,ω−1(f);σG,1(f),ifh=σG,ω2(f);σG,ω2−1(h),otherwise. As in Theorem 2.1, the various expressions must only be applied when h is an edge in the appropriate minor. Proof Immediate from: (a) Theorem 2.1(2,1), (5,1); (b) Theorem 2.1(2,2), (5,2); (c) Theorem 2.1(1,1), (4,1), (5,1); (d) Theorem 2.1(1,2), (5,2); (e) Theorem 2.1(3,1), (5,1); (f) Theorem 2.1(3,2), (4,2), (5,2).□ Lemma 3.5 If f≠σG,ω(e), then  σG[1]e[ω]f,ω=σG[ω]f[1]e,ω. Proof The proof has some similarities to that of Theorem 3.1, but is significantly more complicated. We prove that   σG[1]e[ω]f,ω(g)=σG[ω]f[1]e,ω(g)for all g∈E(G)⧹{e,f}. If g∉{σG,ω−1(e),σG,ω−1(f),σG[1]e,ω−1(f),σG[ω]f,ω−1(e)}, then   σG[1]e[ω]f,ω(g)=σG[1]e,ω(g)=σG,ω(g)=σG[ω]f,ω(g)=σG[ω]f[1]e,ω(g),by Theorem 2.1(5,1), (5,2). This leaves four special cases for g, which we will consider in turn, after noting some facts which we will use repeatedly. Observe that the Lemma’s condition, f≠σG,ω(e), implies   σG[1]e,ω−1(f)≠σG,ω−1(e), (3.3)by Lemma 3.4(a). Also, by Lemma 3.4(b),   e=σG,ω(f)⟺σG[ω]f,ω−1(e)=σG,ω−1(f). (3.4) Case 1: g=σG,ω−1(e).  σG[1]e[ω]f,ω(σG,ω−1(e))=σG[1]e,ω(σG,ω−1(e))(byTheorem2.1(5,2),using(3.3))=σG,ω(e)(byTheorem2.1(2,1)). On the other hand, if e≠σG,ω(f), then   σG[ω]f[1]e,ω(σG,ω−1(e))=σG[ω]f[1]e,ω(σG[ω]f,ω−1(e))(byLemma3.4(b))=σG[ω]f,ω(e)(byTheorem2.1(2,1))=σG,ω(e)(byTheorem2.1(5,2),using(3.3)),while if e=σG,ω(f), then σG,ω−1(e)=f which is not in the domain of σG[ω]f[1]e,ω so this situation does not arise. So, in any event, σG[1]e[ω]f,ω(g)=σG[ω]f[1]e,ω(g) in this case. Case 2: g=σG,ω−1(f).   σG[1]e[ω]f,ω(σG,ω−1(f))=σG[1]e[ω]f,ω(σG[1]e,ω−1(f))(byLemma3.4(a),usingf≠σG,ω(e))=σG[1]e,ω(f)(byTheorem2.1(2,2))={σG,ω(f)(byTheorem2.1(5,1)),iff≠σG,ω−1(e),σG[1]e,ω(σG,ω−1(e)),iff=σG,ω−1(e),={σG,ω(f),iff≠σG,ω−1(e),σG,ω(e),iff=σG,ω−1(e). Now consider σG[ω]f[1]e,ω(σG,ω−1(f)). If e≠σG,ω(f), we have   σG[ω]f[1]e,ω(σG,ω−1(f))=σG[ω]f,ω(σG,ω−1(f))(byTheorem2.1(5,1)and(3.4))=σG,ω(f)(byTheorem2.1(2,2)).If e=σG,ω(f), then   σG[ω]f[1]e,ω(σG,ω−1(f))=σG[ω]f[1]e,ω(σG[ω]f,ω−1(e))(byLemma3.4(b)and(3.4))=σG[ω]f,ω(e)(byTheorem2.1(2,1))=σG,ω(e)(byTheorem2.1(5,2),usinge≠σG,ω−1(f)). Case 3: g=σG[1]e,ω−1(f).  σG[1]e[ω]f,ω(σG[1]e,ω−1(f))=σG[1]e,ω(f)(byTheorem2.1(2,2))={σG,ω(f)(byTheorem2.1(5,1)),ife≠σG,ω(f),σG[1]e,ω(σG,ω−1(e)),ife=σG,ω(f),={σG,ω(f),ife≠σG,ω(f),σG,ω(e),ife=σG,ω(f)(byTheorem2.1(2,1)). If e≠σG,ω(f), then   σG[ω]f[1]e,ω(σG[1]e,ω−1(f))=σG[ω]f[1]e,ω(σG,ω−1(f))(byLemma3.4(a),usingf≠σG,ω(e))=σG[ω]f,ω(σG,ω−1(f))(byTheorem2.1(5,1)and(3.4))=σG,ω(f)(byTheorem2.1(2,2)). If e=σG,ω(f), then   σG[ω]f[1]e,ω(σG[1]e,ω−1(f))=σG[ω]f[1]e,ω(σG,ω−1(f))(byLemma3.4(a),usingf≠σG,ω(e))=σG[ω]f[1]e,ω(σG[ω]f,ω−1(e))(byLemma3.4(b)and(3.4))=σG[ω]f,ω(e)(byTheorem2.1(2,1))=σG,ω(e)(byTheorem2.1(5,2),usinge≠σG,ω−1(f)). Case 4: g=σG[ω]f,ω−1(e). This case can be proved in a manner similar to the previous cases. But in fact this is not necessary, since we have shown the permutations σG[1]e[ω]f,ω and σG[ω]f[1]e,ω agree on every element of their common domain except one, so they must agree on this last element too.□ Lemma 3.6 If f≠σG,ω(e), then  σG[1]e[ω]f,1=σG[ω]f[1]e,1. Proof This proof is more complicated again than that of Lemma 3.5. We may suppose that neither e nor f is a triloop, since we have already established commutativity in such cases, in Lemma 3.2. So σG,μ(e)≠e and σG,μ(f)≠f, for μ∈{1,ω,ω2}. We prove that   σG[1]e[ω]f,1−1(g)=σG[ω]f[1]e,1−1(g) (3.5)for all g∈E(G)⧹{e,f}. Observe that   σG[1]e[ω]f,ω2(g)=σG[1]e,ω2(g)ifg∉{σG[1]e,ω2−1(f),σG[1]e,1(f)},byTheorem2.1(5,2)=σG,ω2(g)ifg≠σG,ω2−1(e),byTheorem2.1(5,1)=σG[ω]f,ω2(g)ifg∉{σG,ω2−1(f),σG,1(f)},byTheorem2.1(5,2)=σG[ω]f[1]e,ω2(g)ifg≠σG[ω]f,ω2−1(e),byTheorem2.1(5,1). It follows that if g∉{σG,1(f),σG,ω2−1(f),σG,ω2−1(e),σG[ω]f,ω2−1(e),σG[1]e,ω2−1(f),σG[1]e,1(f)},then   σG[1]e[ω]f,1−1(g)=σG[1]e[ω]f,ω(σG[1]e[ω]f,ω2(g))(usingσ1◦σω◦σω2=identity)=σG[1]e[ω]f,ω(σG[ω]f[1]e,ω2(g))(bythepreviousparagraph)=σG[ω]f[1]e,ω(σG[ω]f[1]e,ω2(g))(byLemma3.5)=σG[ω]f[1]e,1−1(g)(usingσ1◦σω◦σω2=identity,again). We now consider in turn how to deal with the exceptional values of g, apparently six in number. Case 1: g=σG,1(f). We must have σG,1(f)≠e, else g=e which is forbidden. We have   σG[1]e[ω]f,1−1(σG,1(f))=σG[1]e[ω]f,1−1(σG[1]e,1(f))(byTheorem2.1(5,1),usingf≠σG,ω(e))=σG[1]e,1−1(f)(byLemma3.4(d),firstcase)={σG,ω(e),iff=σG,1(e),σG,1−1(e),iff=σG,ω2−1(e),σG,1−1(f),otherwise,by Lemma 3.4(c). Now consider σG[ω]f[1]e,1−1(σG,1(f)). If f=σG,1(e), then σG[ω]f,1(e)=σG[ω]f,1(σG,1−1(f))=σG,1(f), with the second equality following from Theorem 2.1(1,2). This justifies the first step of the following:   σG[ω]f[1]e,1−1(σG,1(f))=σG[ω]f[1]e,1−1(σG[ω]f,1(e))=σG[ω]f,ω(e)(byLemma3.4(c))=σG,ω(e)(usingourhypothesis,e≠σG,ω−1(f)). If f=σG,ω2−1(e), i.e., e=σG,ω2(f), then σG[ω]f,ω2−1(e)=σG,1(f), by Lemma 3.4(f) (second case). This justifies the first step of the following:   σG[ω]f[1]e,1−1(σG,1(f))=σG[ω]f[1]e,1−1(σG[ω]f,ω2−1(e))=σG[ω]f,1−1(e)(byLemma3.4(c))=σG,1−1(e)(byLemma3.4(d),usinge≠σG,1(f)). Suppose, then, that f≠σG,1(e) and f≠σG,ω2−1(e). From e≠σG,1−1(f), we deduce that σG[ω]f,1(e)=σG,1(e), by Theorem 2.1(5,2). Also, since e≠f and σG,1 is a bijection, we have σG,1(e)≠σG,1(f). So σG[ω]f,1(e)≠σG,1(f). From e≠σG,ω2(f), and our hypothesis e≠σG,ω−1(f), we deduce from Lemma 3.4(f) that σG[ω]f,ω2−1(e)=σG,ω2−1(e). Our hypothesis e≠σG,ω−1(f) implies e≠σG,ω2(σG,1(f)), which in turn implies σG,ω2−1(e)≠σG,1(f). Combining the conclusions of the two previous sentences, we obtain σG[ω]f,ω2−1(e)≠σG,1(f). The conclusions of the previous two paragraphs, together with Lemma 3.4(c), justify the first step in the following:   σG[ω]f[1]e,1−1(σG,1(f))=σG[ω]f,1−1(σG,1(f))=σG,1−1(f)(byLemma3.4(d)). We have shown, then, that σG[1]e[ω]f,1−1 and σG[ω]f[1]e,1−1 agree on g=σG,1(f), in all circumstances. This deals with the first of our exceptional values of g. Case 2: g=σG,ω2−1(f). We must have σG,ω2−1(f)≠e, else g=e which is forbidden. First, observe that σG[1]e,1(f)∈{σG,ω2−1(e),σG,1(f)}, by Theorem 2.1(1,1), (5,1), using the hypothesis f≠σG,ω(e). Now, σG,ω2−1(e)≠σG,ω2−1(f), since e≠f. Furthermore, σG,1(f)≠σG,ω2−1(f), since if σG,1(f)=σG,ω2−1(f), then f=σG,ω2(σG,1(f))=σG,ω−1(f), which means that f is a triloop, which we excluded at the start. So, whatever its value, we have σG[1]e,1(f)≠σG,ω2−1(f). Secondly, observe that σG[1]e,ω2−1(f)=σG,ω2−1(f), by Lemma 3.4(e), using f≠σG,ω2(e) (see the start of this Case). The conclusions of these two previous paragraphs justify the first two steps in the following:   σG[1]e[ω]f,1−1(σG,ω2−1(f))=σG[1]e,1−1(σG,ω2−1(f))(byLemma3.4(d))=σG[1]e,1−1(σG[1]e,ω2−1(f))=σG[1]e,ω(f)={σG,ω(e),iff=σG,ω−1(e),σG,ω(f),otherwise,by Theorem 2.1(2,1), (5,2). Now consider σG[ω]f[1]e,1−1(σG,ω2−1(f)). If f=σG,ω−1(e), then σG,ω2−1(f)=σG,1(e). Also, e≠σG,1−1(f), since if e=σG,1−1(f) then f=σG,ω−1(σG,1−1(f))=σG,ω2(f),so that f is a triloop, which we have excluded. So σG[ω]f,1(e)=σG,1(e)=σG,ω2−1(f), with the first equality holding by Theorem 2.1(5,2). This justifies the first step in the following:   σG[ω]f[1]e,1−1(σG,ω2−1(f))=σG[ω]f[1]e,1−1(σG[ω]f,1(e))=σG[ω]f,ω(e)(byLemma3.4(c))=σG,ω(e)(byTheorem2.1(5,2),sincee≠σG,ω−1(f)byhypothesis). If f≠σG,ω−1(e), then σG,ω2−1(f)≠σG,1(e). Also, σG,ω2−1(f)≠σG,1(f), else f is a triloop, as we saw early in this Case. So σG,ω2−1(f)∉{σG,1(e),σG,1(f)}. But σG[ω]f,1(e)∈{σG,1(e),σG,1(f)}, by Theorem 2.1(1,2), (5,2). So σG,ω2−1(f)≠σG[ω]f,1(e). Since e≠σG,ω−1(f) by hypothesis, σG[ω]f,ω2−1(e)∈{σG,1(f),σG,ω2−1(e)}. Now, as we have seen, σG,ω2−1(f)≠σG,1(f), else f is a triloop; also, σG,ω2−1(f)≠σG,ω2−1(e), since e≠f. So σG,ω2−1(f)≠σG[ω]f,ω2−1(e). The conclusions of the previous two paragraphs, together with Lemma 3.4(c), justify the first step of the following:   σG[ω]f[1]e,1−1(σG,ω2−1(f))=σG[ω]f,1−1(σG,ω2−1(f))=σG,1−1(σG,ω2−1(f))(byLemma3.4(d),usingσG,ω2−1(f)≠σG,1(f))=σG,ω(f). So σG[1]e[ω]f,1−1 and σG[ω]f[1]e,1−1 agree on g=σG,ω2−1(f), in all circumstances. This deals with the second of our exceptional values of g. Case 3: g=σG,ω2−1(e). We must have σG,ω2−1(e)≠f, else g=f which is forbidden. Observe that σG[1]e,1(σG,1−1(e))=σG,ω2−1(e),by Theorem 2.1(1,1). So, if f=σG,1−1(e), then   σG[1]e[ω]f,1−1(σG,ω2−1(e))=σG[1]e[ω]f,1−1(σG[1]e,1(f))=σG[1]e,1−1(f)(byLemma3.4(d))={σG,ω(e),ifalsof=σG,1(e),σG,1−1(f),ifalsof≠σG,1(e),by Lemma 3.4(c) with f≠σG,ω2−1(e). On the other hand, if f≠σG,1−1(e), then σG[1]e,1(f)≠σG[1]e,1(σG,1−1(e)), since σG[1]e,1 is a bijection. So σG,ω2−1(e)≠σG[1]e,1(f). Therefore, we have   σG[1]e[ω]f,1−1(σG,ω2−1(e))=σG[1]e,1−1(σG,ω2−1(e))(byLemma3.4(d))=σG,1−1(e)(byLemma3.4(c)). So, in summary,   σG[1]e[ω]f,1−1(σG,ω2−1(e))={σG,ω(e),iff=σG,1−1(e)andf=σG,1(e),σG,1−1(f),iff=σG,1−1(e)andf≠σG,1(e),σG,1−1(e),iff≠σG,1−1(e). Now consider σG[ω]f[1]e,1−1(σG,ω2−1(e)). Since e≠σG,ω−1(f), by hypothesis, and e≠σG,ω2(f) (see start of this Case), Lemma 3.4(f) gives σG,ω2−1(e)=σG[ω]f,ω2−1(e). Consider, for a moment, the circumstances under which σG[ω]f,1−1(e)=e. Lemma 3.4(d) tells us that   σG[ω]f,1−1(e)={σG,1−1(f),ife=σG,1(f),σG,1−1(e),ife≠σG,1(f).If e≠σG,1(f), then σG[ω]f,1−1(e)≠e, since otherwise e=σG,1−1(e), so that e is a triloop, which we have excluded. Also, if e≠σG,1−1(f), then e cannot equal either of the two possible expressions just given for σG[ω]f,1−1(e) (using the triloop exclusion, again, for the second of these). So, again, σG[ω]f,1−1(e)≠e. On the other hand, if e=σG,1(f) and e=σG,1−1(f), then the first case above gives σG[ω]f,1−1(e)=σG,1−1(f)=e. In this situation, in applying Theorem 2.1 to G[ω]f, we cannot use case (1,1), since that would require σG[ω]f,1−1(e)≠e. Similarly, we cannot use the inverse of case (1,1) to find σG[ω]f[1]e,1−1(σG[ω]f,ω2−1(e)); instead, we must use case (5,1). If e=σG,1(f) and e=σG,1−1(f), then, we have   σG[ω]f[1]e,1−1(σG,ω2−1(e))=σG[ω]f[1]e,1−1(σG[ω]f,ω2−1(e))=σG[ω]f,1−1(σG[ω]f,ω2−1(e))(byTheorem2.1(5,1))=σG[ω]f,ω(e)=σG,ω(e)(byTheorem2.1(5,2),usingourhypothesise≠σG,ω−1(f)). Otherwise, we have   σG[ω]f[1]e,1−1(σG,ω2−1(e))=σG[ω]f[1]e,1−1(σG[ω]f,ω2−1(e))=σG[ω]f,1−1(e)(byLemma3.4(c))={σG,1−1(f),ife=σG,1(f)(andsoe≠σG,1−1(f)too,elseweareinthepreviousparagraph),σG,1−1(e),ife≠σG,1(f),by Lemma 3.4(d). So σG[1]e[ω]f,1−1 and σG[ω]f[1]e,1−1 agree on g=σG,ω2−1(e), in all circumstances. This deals with the third of our exceptional values of g. Cases 4–6: g∈{σG[ω]f,ω2−1(e),σG[1]e,ω2−1(f),σG[1]e,1(f)}. Theorem 2.1 and Lemma 3.4 tell us that   σG[ω]f,ω2−1(e)∈{σG,1(f),σG,ω2−1(e)},σG[1]e,ω2−1(f)∈{σG,ω2−1(e),σG,ω2−1(f)},σG[1]e,1(f)∈{σG,1(f),σG,ω2−1(e)}. So these are not really new cases at all; they each take us back into one of Cases 1–3. This completes our proof of (3.5), and hence of the Lemma.□ Theorem 3.7 If f≠σG,ω(e), then  G[1]e[ω]f=G[ω]f[1]e. Proof In view of Lemmas 3.5 and 3.6, we know that σG[1]e[ω]f,μ=σG[ω]f[1]e,μ for μ∈{1,ω}. But then it follows for μ=ω2 too, since σω2=σω−1◦σ1−1.□ Triality gives the following two corollaries. Corollary 3.8 If f≠σG,1(e), then  G[ω]e[ω2]f=G[ω2]f[ω]e.□ Corollary 3.9 If f≠σG,ω2(e), then  G[ω2]e[1]f=G[1]f[ω2]e.□ The results so far in this section (together with the fact of non-commutativity in general for the excluded cases for the previous three results) give us a complete description of when the μ-reductions do, or do not, commute, in general. But some interesting questions remain. Given that the excluded (generally non-commutative) cases are so specific, it is natural to ask for a characterization of those alternating dimaps for which all reductions always commute. Consider f=σG,ω(e), illustrated in Fig. 5. In this case, •[1]e and •[ω]f do not commute in general, but we can still investigate when they do. Proposition 3.10 If f=σG,ω(e), then •[1]eand •[ω]fcommute if and only if at least one of e, f is a triloop. Proof If either e or f is a triloop, then they commute by Lemma 3.2. Suppose then that neither e nor f is a triloop. If e and f form an a-face of size 2, then it is routine to show that these reductions do not commute unless the head of f meets no other edge except e, or the head of e meets no other edge except f; but that would make either e or f a 1-loop. If e and f do not form such an a-face, then the endpoints of e and f—three in number—are all distinct. The situation is then exactly as in Fig. 5 (except that the right-hand vertex might coincide with the tail of f or the head of e, but that is immaterial). It is evident from the figure that the only way the reductions can commute in this case is if the head of e has in-degree 1 (i.e., if the edges shown in green do not exist), which would make e a 1-loop.□ Theorem 3.11 Every pair of reductions on G commutes if and only if the set of triloops of G includes at least one of each pair of edges that are consecutive in any in-star, a-face or c-face. Proof Use Proposition 3.10 and triality.□ We pause now to introduce a graph derived from G which gives an alternative way of framing Theorem 3.11. The trimedial graph tri(G) of the alternating dimap G has vertex set E(G) with two vertices of tri(G) being adjacent if their corresponding edges in G are consecutive in an a-face, a c-face or an in-star of G. The trimedial graph is an alternating dimap analogue of the medial graph of a classical embedded graph. It is always undirected and 6-regular, and may have loops and/or multiple edges. Its 6-regularity implies that if it has no loops or multiple edges, then it is non-planar even if G is plane (in contrast to the usual medial graph). With this definition, we may rewrite Theorem 3.11. Corollary 3.12 Every pair of reductions on G commutes if and only if the set of triloops of G is a vertex cover of tri(G). So far, we have considered the usual kind of commutativity, where the order in which two operations are applied does not matter. We can also ask about stronger forms of commutativity. If a set of k reductions (each of the form •[μ]e, where each μ∈{1,ω,ω2} and all the e are distinct) has the property that applying them in any order always gives the same result, then we say that it is k-commutative on G. We say that G is k-reduction-commutative if every set of k reductions is k-commutative on G. It is totally reduction-commutative if it is k-reduction-commutative for every k. In this terminology, ordinary commutativity is 2-commutativity, in the sense that, if two particular reductions •[μ]e and •[ν]f commute, then the set {•[μ]e,•[ν]f} is 2-commutative. Theorem 3.11 characterizes alternating dimaps that are 2-reduction-commutative. While total reduction-commutativity implies k-reduction-commutativity for any fixed k, which in turn implies l-reduction-commutativity for any l<k, the converses do not hold. Considering how taking minors affects these properties, we see that, if G is totally reduction-commutative, then so is any minor of G. By contrast, 2-reduction-commutativity is not in general preserved by taking minors. To see this, let H be any alternating dimap with no triloops, and form G from it by inserting an ω2-loop at each vertex of each anticlockwise face and an ω-loop at each vertex of each clockwise face. Then H is a minor of G, yet Theorem 3.11 tells us that G is 2-reduction-commutative yet H is not. We now characterize alternating dimaps that are totally reduction-commutative. A 1-circuit is an alternating dimap consisting of a single directed circuit, in which every edge is a 1-loop. An ω-circuit (respectively, ω2-circuit) consists of a single vertex together with a number of ω-loops (respectively, ω2-loops) at it. A tricircuit is an alternating dimap that can be constructed from a 1-circuit, an ω-circuit and an ω2-circuit (any of which may have no edges), taking a single vertex in each, and identifying these three vertices in the natural way. This is done so as to preserve the alternating dimap property, and will entail having the ω-circuit and ω2-circuit on opposite sides of the 1-circuit. Theorem 3.13 An alternating dimap G is totally reduction-commutative if and only if each of its components is a tricircuit. Proof Suppose G is totally reduction-commutative. Then it is certainly 2-reduction-commutative, so by Theorem 3.11 the set of triloops of G includes at least one of each pair of edges that are consecutive in any in-star, a-face or c-face. Consider those edges of G which have distinct endpoints (i.e., the non-loops). Suppose two non-loop edges e and f share an endpoint v, so e,f∈I(v). Since e and f are not loops, they do not come out of v. The number of half-edges going out of v must be two greater than the number of half-edges other than e and f going into v. So there must be two half-edges going out of v that do not match (that is, are not part of the same edge as) any half-edge going into v. Let g be an edge to which one of these half-edges belongs. Without loss of generality, suppose that e,g,f occur in that order, going clockwise around v. Let the sequence of edges of I(v) which are between g and e going anticlockwise be h1,…,ha, and let the sequence of edges of I(v) which are between g and f going clockwise be i1,…,ic. Then the alternating dimap G′≔G[ω2](h1,…,ha)[ω](i1,…,ic) is left with e,g,f intact, still in this same order around v, and with no edges intervening between them any more. Then e and f are consecutive (clockwise) in the in-star at v in G′. By Theorem 3.11, this implies non-commutativity of some reductions on G′, which in turn implies that G is not totally reduction-commutative. Now suppose two non-triloop non-loops e and f are head-to-tail: say, with v = head of e = tail of f. Since e is not a 1-loop, there must be other edges at v. If all of those edges lie between e and f going clockwise, then e and f are consecutive around the clockwise face containing e, so Theorem 3.11 gives non-commutativity of some reductions, so G is not totally reduction-commutative. Similarly, if those extra edges at v all lie on the other side—between f and e going clockwise—then, again, G is not totally reduction-commutative. So there are some edges on each side. Let the edges of I(v) between f and e going anticlockwise be h1,…,hk. Then G′≔G[ω2](h1,…,hk) has e and f as consecutive edges in the anticlockwise face containing e. This gives some non-commutative reductions in G′, so G is not totally reduction-commutative. If non-triloop non-loops e and f belong to the same component of G, then let P be the shortest path, in the underlying undirected graph, from one to the other. (Note, e,f∉E(P), and P meets e and f only at the endpoints of P, by its minimality.) If all the edges of P are contracted, to give G[1]E(P), then we have e and f sharing an endpoint and we are in one of the previous two paragraphs, so G[1]E(P) is not totally reduction-commutative, so neither is G. So each component of G has at most one edge that is neither a triloop nor a loop. All the 1-loops in a component of G must lie in a single directed circuit in that component. To see this, take any 1-loop e = uv. It has a unique successor, which cannot be a loop or e would not be a 1-loop. So it must either be a 1-loop or the sole edge which is neither a triloop nor a loop. Now let us go back the other way. Consider the edges in I(u). At least one of them must be a non-loop. But if I(u) has two non-loops, then both of them are not 1-loops, and so this component has at least two edges that are neither a triloop nor a loop, which is a contradiction. So I(u) has only one non-loop, which must either be a 1-loop or the sole non-triloop non-loop. We can follow 1-loops forwards and backwards in this way until we are forced to stop. This happens when we complete a circuit, which will either be a circuit consisting entirely of 1-loops—in which case it is an entire component of G—or consisting of 1-loops except for the sole non-triloop non-loop, which we call f=wx. In the latter case, other edges may meet the head x of that special edge, but cannot meet any other vertex on the circuit. The other edges at x must all be loops, since if any is an outgoing non-loop then another must be an incoming non-loop which is then not a 1-loop either, a contradiction with the uniqueness of f. Furthermore, if any edge g at x is a proper 1-semiloop, then we can form a minor, by reduction of any ω-loops or ω2-loops that get in the way, in which f and g form a configuration that allows non-commutativity. So those other edges at x must all be ω-loops or ω2-loops. This description of the component of G, as a circuit whose edges are 1-loops with possibly one exception, and with the head of that exception holding ω-loops and ω2-loops, identifies the component as a tricircuit. So every component of G is in fact a tricircuit. Conversely, if every component of G is a tricircuit, then each component has at most one edge that is not a triloop, so any two reductions on G commute, by Theorem 3.3. Therefore, G is totally reduction-commutative. □ 4. Excluded minors for fixed genus A posy, or k-posy, is an alternating dimap with one vertex, 2k+1 edges (all loops), and two faces. Its genus is k. Up to isomorphism, there is a single 0-posy, a single 1-posy and four 2-posies. The 0-posy is just a single ultraloop. The 1-posy and the four 2-posies are shown in Fig. 6. Figure 6. View largeDownload slide The 1-posy and the four 2-posies. In each posy, the two faces are coloured grey (clockwise) and white (anticlockwise). The first three posies (top row) remain identical under reversal of all edges. The last two are reversals, and mirror images, of each other. Figure 6. View largeDownload slide The 1-posy and the four 2-posies. In each posy, the two faces are coloured grey (clockwise) and white (anticlockwise). The first three posies (top row) remain identical under reversal of all edges. The last two are reversals, and mirror images, of each other. We use posies to give an excluded minor characterization of alternating dimaps of at most a given genus. An analogous result for surface minors and orientably embedded undirected graphs is established in [14]. Theorem 4.1 A non-empty alternating dimap G has genus <kif and only if none of its minors is a disjoint union of posies of total genus k. Proof The forward implication is clear, since every such union of posies has genus k, and reductions never increase the genus. For the reverse implication, we prove by induction on ∣E(G)∣ that every non-empty G has, as a minor, a disjoint union of posies of total genus γ(G). This is true for ∣E(G)∣=1, since then G is an ultraloop, which is the 0-posy. Now suppose it is true for all alternating dimaps of <m edges, where m>1. Let G be any alternating dimap with m edges. Let e∈E(G). Now, G[1]e, G[ω]e and G[ω2]e each have m−1 edges, so by the inductive hypothesis, G[μ]e has as a minor a disjoint union of posies of total genus γ(G[μ]e), for each μ∈{1,ω,ω2}. Such a minor of G[μ]e is also a minor of G, so we see that G has such a minor, for each such μ. If γ(G[μ]e)=γ(G) for any such μ, we are done. So it remains to consider the case where γ(G[μ]e)<γ(G) (in which case γ(G[μ]e)=γ(G)−1) for each μ and each e∈E(G). The condition γ(G[1]e)<γ(G) implies that e is a proper 1-semiloop, so already we know that every edge of G is a loop that does not enclose its own face. Let e be any edge and let v be the vertex at which e is a loop. The condition γ(G[ω2]e)<γ(G) implies that e is also a proper ω-semiloop. Let F be the face on the right side of e, and let F′ be the face on the right side of the left successor e′ of e (see Fig. 7). If faces F and F′ were distinct, then ω-reduction of e would not reduce the genus and e would not be an ω-semiloop. So F=F′. Applying this same reasoning to the next edge (clockwise from e) in the in-star at v (denoted by f in Fig. 7) shows that the face F′ is, in turn, identical to the second face beyond it (denoted by F″), continuing to go clockwise around v. Continuing in this manner we find that every second ‘face’ around v is really just part of one single face. In a similar manner, the condition γ(G[ω]e)<γ(G) implies that e is also a proper ω2-semiloop, and we find that all the ‘faces’ at v which were not accounted for in the previous paragraph (being every other second face around v) are, again, just one single face (but necessarily distinct from the face F we found there). So the component of G that consists of loops at v is a posy. So G itself is just a disjoint union of all these posies, so we are done.□ Figure 7. View largeDownload slide Proof of Theorem 4.1: faces around v when e is a 1-semiloop and an ω-semiloop. Figure 7. View largeDownload slide Proof of Theorem 4.1: faces around v when e is a 1-semiloop and an ω-semiloop. 5. Tutte invariants We now extend the notion of a Tutte invariant to alternating dimaps, and investigate what invariants of this type exist. We define two types of these invariants, in each case attempting to keep the spirit of the recursive definition of the Tutte polynomial. The first seems very general, while the second is about as simple as possible. Definitions An extended Tutte invariant for alternating dimaps is a function F defined on every alternating dimap such that F is invariant under isomorphism, F(emptyalternatingdimap)=1, and there exist w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l such that, for any alternating dimap G, for any ultraloop e of G,   F(G)=wF(G[∗]e), (5.1) for any proper 1-loop e of G,   F(G)=xF(G[1]e), (5.2) for any proper ω-loop e of G,   F(G)=yF(G[ω]e), (5.3) for any proper ω2-loop e of G,   F(G)=zF(G[ω2]e), (5.4) for any proper 1-semiloop e of G,   F(G)=aF(G[1]e)+bF(G[ω]e)+cF(G[ω2]e), (5.5) for any proper ω-semiloop e of G,   F(G)=dF(G[1]e)+eF(G[ω]e)+fF(G[ω2]e), (5.6) for any proper ω2-semiloop e of G,   F(G)=gF(G[1]e)+hF(G[ω]e)+iF(G[ω2]e), (5.7) for any other edge e of G (i.e., one that is not a semiloop),   F(G)=jF(G[1]e)+kF(G[ω]e)+lF(G[ω2]e). (5.8) A simple Tutte invariant for alternating dimaps is an extended Tutte invariant for which a=b=c=d=e=f=g=h=i=j=k=l=1. In other words, proper semiloops are all treated the same as each other, and as non-semiloops, and all the coefficients in their recursive expressions are 1. We first describe all simple Tutte invariants. Theorem 5.1 The only simple Tutte invariants of alternating dimaps are the following: F(G)=0for non-empty G, with w=0; F(G)=3∣E(G)∣, with w=x=y=z=3; F(G)=(−1)∣V(G)∣, with y=z=1and x=w=−1; F(G)=(−1)af(G), with x=z=1and y=w=−1; F(G)=(−1)cf(G), with x=y=1and z=w=−1. Proof Let F be a simple Tutte invariant of alternating dimaps, with w,x,y,z as in the definition. If w=0, then it is easily shown that F(G)=0 for all non-empty G. So suppose w≠0. For any k≥1, let Uk be a disjoint union of k ultraloops. For any k≥2, let Lk,1 be the directed cycle of k vertices and k edges, which has one a-face, one c-face and k in-stars. For k≥2 and μ∈{ω,ω2}, let Lk,μ be the alternating dimap consisting of a single vertex with kμ-loops. It is clear that   F(Uk)=wk, (5.9)  F(L2,1)=xw, (5.10)  F(L2,ω)=yw, (5.11)  F(L2,ω2)=zw. (5.12) Consider the alternating dimap L2,1+eω2 obtained by adding an ω2-loop e to a vertex v of L2,1. (This is the black dimap in Fig. 1(b).) Let f (respectively, g) be the edge of L2,1 going out of (respectively, into) v. Observe that f is a 1-loop and g is not a triloop. We can calculate F(L2,1+eω2) by applying (5.2) at f (or (5.4) at e), obtaining xzw. Alternatively, we can apply (5.8) at g, obtaining (z+x+w)w. Equating the results and using w≠0, we obtain   x+z+w=xz. (5.13)Similar reasoning for the trials (L2,1+eω2)ω and (L2,1+eω2)ω2 gives   F((L2,1+eω2)ω)=xyw, (5.14)  F((L2,1+eω2)ω2)=yzw, (5.15)  x+y+w=xy, (5.16)  y+z+w=yz. (5.17) Now consider the alternating dimap obtained from L2,1 (with edges g,h) by adding, to the endpoint v of h, an ω2-loop e within the anticlockwise face and an ω-loop f within the clockwise face. Call it A. Using the ω2-loop e, or the ω-loop f, or the 1-loop g, we find that   F(A)=xyzw. (5.18)But using h, which is not a triloop, we have   F(A)=F((L2,1+eω2)ω2)+F((L2,1+eω2)ω)+F(L2,1+eω2)=yzw+xyw+xzw=(yz+xy+xz)w.Equating with (5.18), and using w≠0, we obtain   xz+xy+yz=xyz. (5.19) From (5.13), (5.16) and (5.17), we obtain   w=xz−x−z=xy−x−y=yz−y−z.The second equality here gives (x−1)z=(x−1)y, so either x=1 or y=z. Similarly, either y=1 or x=z, and either z=1 or x=y. Combining these, we have one of x=y=z, x=z=1 and y≠1, x=y=1 and z≠1, y=z=1 and x≠1.If (i) holds, then any of (5.13), (5.16) and (5.17) gives   w=x(x−2). (5.20)Also (5.19) gives 3x2=x3, whence x=3 (since x=0 would imply w=0, by (5.20)) and w=3 (by (5.20)). If (ii) holds, then (5.16) gives w=−1. Similarly, cases (iii) and (iv) give w=−1 too. Also, (5.19) implies y=−1 in case (ii), z=−1 in case (iii) and x=−1 in case (iv). We now establish the form of F for each of cases (i)–(iv) in turn. The numbering of the claims indicates the case to which each applies. Claim (i): F(G)=3∣E(G)∣. Proof of Claim (i): we use induction on ∣E(G)∣. If ∣E(G)∣=0 then the claim is true by the definition of F. Suppose ∣E(G)∣=m>1. Let e∈E(G). If e is an ultraloop, then F(G)=wF(G[∗]e)=3F(G[∗]e)=3·3m−1, by the inductive hypothesis, which equals 3m. If e is a proper μ-loop, with μ∈{1,ω,ω2}, then F(G)=xF(G[μ]e)=3·3m−1=3m. If e is not a triloop, then F(G)=F(G[1]e)+F(G[ω]e)+F(G[ω2]e)=3m−1+3m−1+3m−1=3m. Claim (ii): F(G)=(−1)af(G). Proof of Claim (ii): we use induction on ∣E(G)∣. If ∣E(G)∣=0, then the claim is true by the definition of F. Suppose ∣E(G)∣=m>1. Let e∈E(G). If e is an ultraloop, then F(G)=wF(G[∗]e)=−F(G[∗]e)=−(−1)af(G)−1=(−1)af(G). Observe that the number of anticlockwise faces in an alternating dimap goes down by 1 when an ω-loop is reduced, and it may be altered when an edge is ω2-reduced. But if e is not an ω-loop, then the number of anticlockwise faces is unchanged by 1- or ω-reduction. If e is a proper ω-loop, then F(G)=yF(G[ω]e)=−F(G[ω]e)=−(−1)af(G[ω]e)=−(−1)af(G)−1=(−1)af(G).If e is a proper μ-loop with μ∈{1,ω2}, then F(G)=F(G[μ]e)=(−1)af(G[μ]e)=(−1)af(G). If e is a proper ω-semiloop, then af(G[ω2]e)=af(G)+1, while if e is neither a triloop nor an ω-semiloop, then af(G[ω2]e)=af(G)−1.In any event, if e is not a triloop, then F(G)=F(G[1]e)+F(G[ω]e)+F(G[ω2]e)=(−1)af(G[1]e)+(−1)af(G[ω]e)+(−1)af(G[ω2]e)=(−1)af(G)+(−1)af(G)+(−1)af(G)±1=(−1)af(G). Claim (iii): F(G)=(−1)cf(G). Claim (iv): F(G)=(−1)∣V(G)∣. The proofs of Claims (iii) and (iv) are similar to that of Claim (ii), and are left as an exercise. For Claim (iv), bear in mind that ∣V(G)∣ is the number of in-stars of G.□ We now turn to extended Tutte invariants. A basic one is   F(G)=α∣E(G)∣β∣V(G)∣γaf(G)δcf(G),with α,β,γ,δ≠0. This satisfies the definition with w=αβγδ, x=αβ, y=αγ, z=αδ, a=α/β, f=α/δ, h=α/γ, b=c=d=e=g=i=0, j=αβ/3, k=αγ/3, and l=αδ/3. Extended Tutte invariants are much richer than simple Tutte invariants, since they include the Tutte polynomial for planar graphs, in a sense we now explain. The Tutte polynomial T(G;x,y) of a graph G has the following inductive definition. If E(G)=∅ then T(G;x,y)=1. Otherwise, for any e∈E(G),   T(G;x,y)={xT(G⧹e;x,y),ifeisacoloop;yT(G/e;x,y),ifeisaloop;T(G/e;x,y)+T(G⧹e;x,y),otherwise. To any orientably 2-cell-embedded (undirected) graph G, we can associate two alternating dimaps altc(G) and alta(G) as follows. For altc(G) (respectively, alta(G)), replace each edge e=uv∈E(G) by a pair of oppositely directed edges (u,v) and (v,u), forming a clockwise (respectively, anticlockwise) face of size two. The faces of G now all correspond to anticlockwise (respectively, clockwise) faces in altc(G) (respectively, alta(G)). For any alternating dimap G, define Tc(G;x,y) and Ta(G;x,y) as follows. If E(G)=∅, then Tc(G;x,y)=Ta(G;x,y)=1. Otherwise, for any e∈E(G),   Tc(G;x,y)={Tc(G[∗]e;x,y),ifeisanω2-loop(includinganultraloop);xTc(G[ω2]e;x,y),ifeisanω-semiloop;yTc(G[1]e;x,y),ifeisaproper1-semilooporanω-loop;Tc(G[1]e;x,y),+Tc(G[ω2]e;x,y),ifeisnotasemiloop.Ta(G;x,y)={Ta(G[*]e;x,y),ifeisanω-loop(includinganultraloop);xTa(G[ω]e;x,y),ifeisanω2-semiloop;yTa(G[1]e;x,y),ifeisaproper1-semilooporanω2-loop;Ta(G[1]e;x,y),+Ta(G[ω]e;x,y),ifeisnotasemiloop. Theorem 5.2 For any plane graph G,   T(G;x,y)=Tc(altc(G);x,y)=Ta(alta(G);x,y). Proof For any vertex v, write L(ω)(v) and L(ω2)(v) for an ω-loop and an ω2-loop, respectively, at v. If such a loop is added to an alternating dimap, it must be placed within a c-face or an a-face, respectively. Consider altc(G). Observe that, for any uv∈E(G),   altc(G)[1](u,v)=altc(G/uv)+L(ω2)(u′),altc(G)[ω2](u,v)=altc(G⧹uv)+L(ω2)(u′),for some u′. (Mostly u′=u, except that a little more detail is needed if (u,v) is a proper 1-semiloop, but the exact location of these extra triloops is not important.) We use these observations to prove, by induction on ∣E(G)∣, that T(G;x,y)=Tc(altc(G);x,y) for any plane graph G. It is clear from the definitions that they are identical when G is empty. Suppose then that T(G;x,y)=Tc(altc(G);x,y) when ∣E(G)∣<m, where m≥1. Let G be any plane graph on m edges, and let e=uv∈E(G). If e is a coloop, then (u,v) and (v,u) are both ω-semiloops in altc(G). (Conversely, if (u,v) and (v,u) are both proper ω-semiloops in altc(G), then uv is a coloop in G. This does not hold in general if G is not plane, however.) We have   Tc(altc(G);x,y)=xTc(altc(G)[ω2](u,v);x,y)=xTc(altc(G⧹uv)+L(ω2)(u′);x,y)=xTc(altc(G⧹uv);x,y)=xT(G⧹uv;x,y)=T(G;x,y),where the penultimate equality uses the inductive hypothesis. If e is a loop, then in altc(G) the directed versions (u,v) and (v,u) are both 1-semiloops. (This time, the converse holds even if G is not plane.) One of them may also be an ω-loop, but neither is an ω2-loop. In any case, we find that Tc(altc(G);x,y)=T(G;x,y) by a similar argument to that just used for coloops. If e is neither a coloop nor a loop, we have   Tc(altc(G);x,y)=Tc(altc(G)[1](u,v);x,y)+Tc(altc(G)[ω2](u,v);x,y)=Tc(altc(G/uv)+L(ω2)(u′);x,y)+Tc(altc(G⧹uv)+L(ω2)(u′);x,y)=Tc(altc(G/uv);x,y)+Tc(altc(G⧹uv);x,y)=T(G/uv;x,y)+T(G⧹uv;x,y)(bytheinductivehypothesis)=T(G;x,y). We conclude by induction that T(G;x,y)=Tc(altc(G);x,y) holds for all G. The proof that T(G;x,y)=Ta(alta(G);x,y) follows the same line, with appropriate adjustments.□ Having constructed alternating dimaps from embedded graphs, by replacing edges by c-faces, or by a-faces, of size 2, it is natural to ask about replacing edges by in-stars of size 2. To do this, for an embedded graph G, first construct its medial graph, med(G), then turn it into an alternating dimap by directing the edges so as to ensure the alternating property. For each component of G, there are two such ways of directing the edges in that component. There are, therefore, 2k(G) different alternating dimaps constructible from G in this way, all with med(G) as the underlying embedded graph. We refer to any one of them as alti(G). Write Ti(G;x) for any invariant of alternating dimaps that satisfies the following:   Ti(G;x)={1,ifGisempty;Ti(G[*]e;x),ifeisa1-loop(includinganultraloop);xTi(G[ω2]e;x),ifeisaproperω-semilooporanω2-loop;xTi(G[ω]e;x),ifeisaproperω2-semilooporanω-loop;Ti(G[ω]e;x)+Ti(G[ω2]e;x),ifeisnotasemiloop.This is not a full definition of a unique Ti(G;x), since we have not specified what happens if e is a proper 1-semiloop. But, since med(G) is 4-regular, alti(G) has no proper 1-semiloops. Furthermore, the only minors of it we need to form do not require 1-reduction, so these minors are each 4-regular and so have no proper 1-semiloop too. Theorem 5.3 For any plane graph G,   T(G;x,x)=Ti(alti(G);x). Proof For any alternating dimap H, write H(1) for any alternating dimap obtained from H by either adjoining an ultraloop (which becomes its own new component) or subdividing some edge (by insertion in it of a new vertex of indegree=outdegree=1, with the edge going into the new vertex being a proper 1-loop). In either case, a new 1-loop is created. Let G be a plane graph and fix any specific alti(G). If e∈E(G), write e↓ for either of the edges of alti(G) that are directed into the vertex representing e. Observe that if e∈E(G) is neither a loop nor a coloop,   {alti(G)[ω]e↓,alti(G)[ω2]e↓}={alti(G/e)(1),alti(G⧹e)(1)}.Therefore,   Ti(alti(G)[ω]e↓;x)+Ti(alti(G)[ω2]e↓;x)=Ti(alti(G/e)(1);x)+Ti(alti(G⧹e)(1);x).If e is either a coloop or a loop, then   alti(G)[ω2]e↓∈{alti(G/e)(1),alti(G⧹e)(1)},alti(G)[ω]e↓∈{alti(G/e)(1),alti(G⧹e)(1)}. We now prove the theorem by induction on ∣E(G)∣. The base case is immediate from the definition. So suppose G is a plane graph with m edges, where m≥1. If e is a coloop or a loop, then e↓ is an ω- or an ω2-semiloop in alti(G), except that it is not a 1-loop. From our (partial) definition of Ti(alti(G);x), we have   Ti(alti(G);x)∈{xTi(alti(G)[ω]e↓;x),xTi(alti(G)[ω2]e↓;x)}⊆{xTi(alti(G/e)(1);x),xTi(alti(G⧹e)(1);x)}={xTi(alti(G/e);x),xTi(alti(G⧹e);x)}={xT(G/e;x,x),xT(G⧹e;x,x)},by the inductive hypothesis. But, for such an e, the graphs G/e and G⧹e have isomorphic cycle matroids, so their Tutte polynomials are identical. Therefore,   Ti(alti(G);x)=xT(G/e;x,x)=xT(G⧹e;x,x).But these two quantities each equal T(G;x,x), for such an e, so we are done in this case. If e is neither a loop nor a coloop, then   Ti(alti(G);x)=Ti(alti(G)[ω]e↓;x)+Ti(alti(G)[ω2]e↓;x)=Ti(alti(G/e)(1);x)+Ti(alti(G⧹e)(1);x)=Ti(alti(G/e);x)+Ti(alti(G⧹e);x)=T(G/e;x,x)+T(G⧹e;x,x)(bytheinductivehypothesis)=T(G;x,x). The result follows.□ The Tutte polynomial evaluation T(G;x,x) is just the Martin polynomial of med(G) (see for example [20]). 6. Future work This work suggests many questions and problems for further research. Other combinatorial structures with triality and three minor operations As mentioned in Section 1.1, we have determined the relationship between alternating dimaps and binary functions [27]. It remains to determine the precise relationship between alternating dimaps, or indeed binary functions, and the other object types listed there. What do all the permutations generate? The mappings σH,μ, where H ranges over all minors of an alternating dimap G and μ∈{1,ω,ω2}, together generate (under composition) an inverse semigroup, which we denote by IS(G). This suggests the problem of describing IS(G) and classifying it among known types of inverse semigroup. Non-commutative minors in general Are there other types of combinatorial objects with natural minor operations that do not commute? To be convincing, such object types would need to have some of the hallmarks of a good theory of minors, such as excluded minor characterizations and Tutte invariants. Commutativity up to isomorphism So far, we have considered commutativity (or otherwise) with respect to identity: reductions commute if and only if carrying them out in each possible order gives alternating dimaps that are identical. We could also study commutativity with respect to isomorphism, and ask for a characterization of alternating dimaps G for which, for all μ1,μ2∈{1,ω,ω2} and all e,f∈E(G),   G[μ1]e[μ2]f≅G[μ2]f[μ1]e. Other excluded minor characterizations We have given a first excluded-minor result for alternating dimaps. It would be interesting to identify other natural classes of alternating dimaps that are closed under minors and find excluded minor characterizations for them. Antichains under minor inclusion Do all minor-closed classes of alternating dimaps have a finite set of excluded minors? In other words, are alternating dimaps well-quasi-ordered under the minor relation? Extended Tutte invariants We gave a full description of all simple Tutte invariants, but have not done so for extended Tutte invariants. The latter are much richer than the former, since they include the usual Tutte polynomial of an undirected abstract planar graph. But we do not yet know if they contain interesting information that takes the embedding of the dimap into account. Tutte invariants for ordered alternating dimaps One reason that Tutte invariants of alternating dimaps are more limited than Tutte invariants for graphs is the non-commutativity of the minor operations. The definitions of such invariants for alternating dimaps require the stated recursive relations to hold for reduction of any edge of the stated type, which means that the invariant will need to be unperturbable by some variations of the order of operations. These observations raise the possibility that better invariants may come from including an ordering of the edges in the object to which the invariant applies. Definitions An ordered alternating dimap is a pair (G,<), where G is an alternating dimap and < is a linear order on E(G). If (G,<) is an ordered alternating dimap and μ∈{1,ω,ω2}, then the μ-reduction (G,<)[μ] of (G,<) is the ordered alternating dimap (G[μ]e0,<′), where e0 is the first edge in E(G) under < and the order <′ on E(G)⧹{e0} is obtained by simply removing e0 from the order <. Tutte invariants and extended Tutte invariants are defined for these objects by modifying the definitions of such invariants for ordinary alternating dimaps as follows: The definitions apply to ordered alternating dimaps, rather than just to alternating dimaps. All references to G[μ]e are replaced by (G,<)[μ], for each μ. All universal quantification over edges is deleted (since there is no choice of which edge to reduce, since it is always the first edge in the ordering which must be reduced). All reference to an edge e is replaced by reference to the first edge e0 in the ordering.For example, the second condition in each of the definitions becomes: if e0 is a 1-loop, then F((G,<))=xF((G,<)[1]). When G is a general plane alternating dimap, the extended Tutte invariants Tc(G;x,y) and Ta(G;x,y) we considered earlier actually depend on the order in which the edges are considered. So they pertain to ordered alternating dimaps. But, if G has the form altc(H) (with analogous remarks applying to alta(H)), then the order in which the edges uv of H are considered does not matter, and each time a corresponding (u,v) is reduced in G, it leaves behind an ω2-loop which can be reduced at any time. (Note also that if we do not use ω-reductions, we cannot encounter those situations of non-commutativity for two edges that are consecutive in an a-face or an in-star.) For such cases, the invariants are well-defined without having to specify an order on the edges at the beginning. We ask for a characterization of (a) simple Tutte invariants, and (b) extended Tutte invariants, of ordered alternating dimaps. Funding and presentations Part of the work of this paper was done while the author was a Visiting Fellow (Combinatorics and Statistical Mechanics programme) at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January–February 2008, and on sabbatical at the Department of Mathematics and Statistics, University of Melbourne (January–June 2011). 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The Quarterly Journal of MathematicsOxford University Press

Published: Mar 1, 2018

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