Max Point-Tolerance Graphs

TL;DR

This paper introduces max point-tolerance (MPT) graphs, characterizes their properties, relates them to known geometric graph classes, and analyzes key optimization problems on them, revealing their computational complexity and relationships.

Contribution

It formally defines MPT graphs, provides multiple characterizations, and studies algorithmic problems, establishing their position among geometric and classical graph classes.

Findings

MPT graphs can be characterized as intersections of specific 2D geometric graphs.

Maximum weight independent set on MPT graphs is solvable in polynomial time.

Coloring is NP-complete, and clique cover admits a 2-approximation.

Abstract

A graph $G$ is a \emph{max point-tolerance (MPT)} graph if each vertex $v$ of $G$ can be mapped to a \emph{pointed-interval} $(I_v, p_v)$ where $I_v$ is an interval of $\mathbb{R}$ and $p_v \in I_v$ such that $uv$ is an edge of $G$ iff $I_u \cap I_v \supseteq \{p_u, p_v\}$. MPT graphs model relationships among DNA fragments in genome-wide association studies as well as basic transmission problems in telecommunications. We formally introduce this graph class, characterize it, study combinatorial optimization problems on it, and relate it to several well known graph classes. We characterize MPT graphs as a special case of several 2D geometric intersection graphs; namely, triangle, rectangle, L-shape, and line segment intersection graphs. We further characterize MPT as having certain linear orders on their vertex set. Our last characterization is that MPT graphs are precisely obtained by intersecting special pairs of interval graphs. We also show that, on MPT graphs, the maximum weight independent set problem can be solved in polynomial time, the coloring problem is NP-complete, and the clique cover problem has a 2-approximation. Finally, we demonstrate several connections to known graph classes; e.g., MPT graphs strictly contain interval graphs and outerplanar graphs, but are incomparable to permutation, chordal, and planar graphs.