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 mots 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 have an (h,s,t)-representation is denoted [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. 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 Split∩VPT. Additionally, we present a non-trivial subclass of Split∩VPT in which these problems are polynomial time solvable.
@article{arxiv.1112.3254,
title = {Recognizing [h,2,1] graphs},
author = {Liliana Alcón and Marisa Gutierrez and María Pía Mazzoleni},
journal= {arXiv preprint arXiv:1112.3254},
year = {2011}
}