Discrete Mathematics 308 (2008) 1381 – 1387
Representing edge intersection graphs of paths on degree 4 trees
Martin Charles Golumbic
, Marina Lipshteyn
, Michal Stern
Caesarea Rothschild Institute, University of Haifa, Israel
The Academic College of Tel-Aviv, Jaffa, Tel-Aviv, Israel
Received 1 November 2005; received in revised form 10 May 2006; accepted 11 July 2007
Available online 4 September 2007
Let P be a collection of nontrivial simple paths on a host tree T. The edge intersection graph of P, denoted by EPT(P), has vertex
set that corresponds to the members of P, and two vertices are joined by an edge if and only if the corresponding members of P
share at least one common edge in T. An undirected graph G is called an edge intersection graph of paths in a tree if G = EPT(P)
for some P and T. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree network or assigning
wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph.
An undirected graph G is chordal if every cycle in G of length greater than 3 possesses a chord. Chordal graphs correspond to
vertex intersection graphs of subtrees on a tree. An undirected graph G is weakly chordal if every cycle of length greater than 4 in G
and in its complement
G possesses a chord. It is known that the EPT graphs restricted to host trees of vertex degree 3 are precisely
the chordal EPT graphs. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host
tree restricted to degree 4. Moreover, this provides an algorithm to reduce a given EPT representation of a weakly chordal EPT graph
to an EPT representation on a degree 4 tree. Finally, we raise a number of intriguing open questions regarding related families of
© 2007 Elsevier B.V. All rights reserved.
Keywords: Paths of a tree; Intersection graphs; Weakly chordal graphs; EPT graphs; Coloring
Let P be a collection of nontrivial simple paths on a host tree T. We deﬁne two different types of intersection graphs
from the pair P,T, namely the VPT and EPT graphs. We deﬁne the edge intersection graph EPT(P) of P to have
vertices which correspond to the members of P, such that two vertices are adjacent in EPT(P) if and only if the
corresponding paths in P share at least one common edge in T. An undirected graph G is called an edge intersection
graph of paths in a tree (EPT) if G = EPT(P) for some P and T, and we call P,T an EPT representation of G. The
EPT graphs were introduced by Golumbic and Jamison [8,9]. Similarly, the vertex intersection graph VPT(P) of P
has vertices which correspond to the members of P, such that two vertices are adjacent in VPT(P) if and only if the
corresponding paths in P share at least one vertex in T. An undirected graph G is called a vertex intersection graph of
paths in a tree (VPT) if G =VPT(P) for some P and T, and we call P,T a VPT representation of G. In the literature,
VPT graphs are also called
path graphs [6,7].
E-mail address: email@example.com (M. Lipshteyn).
0012-365X/$ - see front matter © 2007 Elsevier B.V. All rights reserved.