Recognizing vertex intersection graphs of paths on bounded degree trees

An (h,s,t)-representation of a graph G consists of a collection of subtrees of a tree T, where each subtree corresponds to a vertex of G such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T. The class of graphs that has an (h,s,t)-representation is denoted by [h,s,t]. An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. Thus, [h,2,1] graphs are the VPT graphs that can be represented in a tree with maximum degree at most h. In this paper we characterize [h,2,1] graphs using chromatic number. We show that the problem of deciding whether a given VPT graph belongs to [h,2,1] is NP-complete, while the problem of deciding whether the graph belongs to [h,2,1]−[h−1,2,1] is NP-hard. Both problems remain hard even when restricted to VPT∩Split. Additionally, we present a non-trivial subclass of VPT∩Split in which these problems are polynomial time solvable.