Intersection graphs of paths in a tree
Journal of Combinatorial Theory Series B
A faster algorithm to recognize undirected path graphs
Discrete Applied Mathematics
Tolerance intersection graphs on binary trees with constant tolerance
Discrete Mathematics
Graphs and Hypergraphs
Tree representations of graphs
European Journal of Combinatorics
The k-edge intersection graphs of paths in a tree
Discrete Applied Mathematics
Equivalences and the complete hierarchy of intersection graphs of paths in a tree
Discrete Applied Mathematics
What Is between Chordal and Weakly Chordal Graphs?
Graph-Theoretic Concepts in Computer Science
Intersection models of weakly chordal graphs
Discrete Applied Mathematics
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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.