On Pairwise Compatibility of Some Graph (Super)Classes

TL;DR

This paper investigates the class of pairwise compatibility graphs (PCGs), proving inclusion of certain subclasses like threshold tolerance graphs and exclusion of others such as disk, grid, and intersection graphs, clarifying their relationships.

Contribution

It establishes new inclusions and non-inclusions among various graph classes within the PCG framework, enhancing understanding of their structural properties.

Findings

Threshold tolerance graphs are included in PCGs.

Disk and grid intersection graphs are not in PCGs.

Some superclasses like trapezoid and permutation graphs are excluded from PCGs.

Abstract

A graph G=(V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers `d' and `D' such that each leaf `u' of T is a node of V and the edge `(u,v) belongs to E' iff `d <= d_T(u, v) <= D' where d_T(u, v) is the sum of weights of the edges on the unique path from `u' to `v' in T. The main issue on these graphs consists in characterizing them. In this note we prove the inclusion in the PCG class of threshold tolerance graphs and the non-inclusion of a number of intersection graphs, such as disk and grid intersection graphs, circular arc and tolerance graphs. The non-inclusion of some superclasses (trapezoid, permutation and rectangle intersection graphs) follows.