A Faster Algorithm to Recognize Undirected Path Graphs

Let F be a nite 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 6 = j) have a non-empty 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 Discr. Math. 23(1978), 211{227] gave the rst 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.