Triangulated edge intersection graphs of paths in a tree

Let T = (V(T), E(T)) be a tree and B = {Pi} denote a collection of non-trivial (i.e., of length at least 1) simple paths in T, where a path P= (v,, u2, . . . , Q) is considered in the sequel as the collection {{uJ, {Q}, . . . ,{z)J, {u,, u2}, {u2, I+}, _ . . , {w,-,, Q}}. The intersection graph 0(S, 9) of 9 over set S has vertices which correspond to the paths in 9 and two vertices uk and U! are adjacent if the corresponding paths Pk and Pl intersect over S, that is, if Pk n Pl rl S = $3. Two kinds of intersection graphs of 9 have been considered so far, for S = V(T) and S = E(T), where in the former case the vertex set of T is considered as the collection of one-element sets. A graph G is a vertex intersection graph of paths in a tree (shortly, VPT graph) if G = f2( V(T), 9) for a certain tree T and a path collection 9? in T, and G is called an edge intersection graph of paths in a tree (EPT graph) if G = O(E(T), 9) for some path collection 9 in a tree T. Neither of these two classes of graphs is contained in the other. Every cycle C,( k 2 4) is an EPT graph but no C,( k 2 4) is a VPT graph. Some examples of VPT graphs which are not EPT graphs can be found in [8] and [4]. The VPT graphs were defined and completely characterized in [6] and the algorithm for their recognition appeared in [l]. The EPT graphs have been introduced and partially characterized in [7] and [8]. Recently, it has been shown that the recognition problem for EPT graphs is NP-complete (see [4]), hence it is unlikely that there exists an elegant and complete characterization of these graphs. The main purpose of this paper is to clarify the status of triangulated EPT graphs, where a graph G is triangulated if every cycle of length at least 4 has a chord, that is, an edge joining two nonconsecutive vertices of the cycle. It was shown in [8] that if G is a triangulated EPT graph then for every vertex u of G, the subgraph of G generated by N(v), the neighbor set of u, is an interval graph. Based on this fact, it was conjectured that every triangulated EPT graph is a VPT graph. Here, we prove this conjecture. The graphs which are both VPT and EPT have been characterized in [4] (see Lemma 1); however, the conjecture does not follow from this result.