A faster algorithm to recognize undirected path graphs

Abstract

Let F be a finite family of nonempty sets. The undirected graph G is called the intersection graph of F if there is a bijection between the members of F and the vertices of G such that any two sets F i and F j (for i≠ j) have a nonempty intersection if and only if the corresponding vertices are adjacent. We study intersection graphs where F is a family of undirected paths in an unrooted, undirected tree; these graphs are called ( undirected) path graphs. They constitute a proper subclass of the chordal graphs. Gavril [Discrete Math. 23 (1978) 211–227] gave the first polynomial time algorithm to recognize undirected path graphs; his algorithm runs in time O( n 4), where n is the number of vertices. The topic of this paper is a new recognition algorithm that runs in time O( mn), where m is the number of edges.