Recognizing [h,2,1] graphs

TL;DR

This paper characterizes a specific class of graphs called [h,2,1] using chromatic number, proves NP-completeness and NP-hardness of related recognition problems, and identifies a polynomial-time solvable subclass.

Contribution

It provides a chromatic number characterization of [h,2,1] graphs and analyzes the computational complexity of recognizing these graphs within VPT and split graph subclasses.

Findings

Recognition of [h,2,1] graphs is NP-complete.

Recognition of [h,2,1]-[h-1,2,1] graphs is NP-hard.

A subclass of split VPT graphs allows polynomial-time recognition.

Abstract

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 $Split \cap VPT$. Additionally, we present a non-trivial subclass of $Split \cap VPT$ in which these problems are polynomial time solvable.