TY - JOUR
AU1 - Ziedan, Emile
AU2 - Rajendraprasad, Deepak
AU3 - Mathew, Rogers
AU4 - Golumbic, Martin
AU5 - Dusart, Jérémie
AB - A linear ordering of the vertices of a graph G
separates two edges of G if both the endpoints of one precede both the endpoints of the other in the order. We call two edges
$$\{a,b\}$$
{
a
,
b
}
and
$$\{c,d\}$$
{
c
,
d
}
of G
strongly independent if the set of endpoints
$$\{a,b,c,d\}$$
{
a
,
b
,
c
,
d
}
induces a
$$2K_2$$
2
K
2
in G. The induced separation dimension of a graph G is the smallest cardinality of a family
$$\mathcal {L}$$
L
of linear orders of V(G) such that every pair of strongly independent edges in G are separated in at least one of the linear orders in
$$\mathcal {L}$$
L
. For each
$$k \in \mathbb {N}$$
k
∈
N
, the family of graphs with induced separation dimension at most k is denoted by
$${\text {ISD}}(k)$$
ISD
(
k
)
. In this article, we initiate a study of this new dimensional parameter. The class
$${\text {ISD}}(1)$$
ISD
(
1
)
or, equivalently, the family of graphs which can be embedded on a line so that every pair of strongly independent edges are disjoint line segments, is already an interesting case. On the positive side, we give characterizations for chordal graphs in
$${\text {ISD}}(1)$$
ISD
(
1
)
which immediately lead to a polynomial time algorithm which determines the induced separation dimension of chordal graphs. On the negative side, we show that the recognition problem for
$${\text {ISD}}(1)$$
ISD
(
1
)
is NP-complete for general graphs. Nevertheless, we show that the maximum induced matching problem can be solved efficiently in
$${\text {ISD}}(1)$$
ISD
(
1
)
. We then briefly study
$${\text {ISD}}(2)$$
ISD
(
2
)
and show that it contains many important graph classes like outerplanar graphs, chordal graphs, circular arc graphs and polygon-circle graphs. Finally, we describe two techniques to construct graphs with large induced separation dimension. The first one is used to show that the maximum induced separation dimension of a graph on n vertices is
$$\Theta (\lg n)$$
Θ
(
lg
n
)
and the second one is used to construct AT-free graphs with arbitrarily large induced separation dimension. The second construction is also used to show that, for every
$$k \ge 2$$
k
≥
2
, the recognition problem for
$${\text {ISD}}(k)$$
ISD
(
k
)
is NP-complete even on AT-free graphs.
TI - The Induced Separation Dimension of a Graph
JF - Algorithmica
DO - 10.1007/s00453-017-0353-x
DA - 2017-07-31
UR - https://www.deepdyve.com/lp/springer-journals/the-induced-separation-dimension-of-a-graph-0NDQ98FLI9
SP - 2834
EP - 2848
VL - 80
IS - 10
DP - DeepDyve