A class of perfectly contractile graphs

TL;DR

This paper proves that graphs in a specific class without certain subgraphs always contain an even pair, and provides a polynomial-time algorithm to find and contract such pairs for optimal coloring.

Contribution

It establishes the existence of even pairs in class ${ mf A}$ graphs and offers a polynomial algorithm for their identification and contraction, enabling optimal coloring.

Findings

Every graph in class ${ mf A}$ has an even pair.

A polynomial-time algorithm finds an even pair that preserves class ${ mf A}$.

This leads to polynomial-time optimal coloring of these graphs.

Abstract

We consider the class ${\cal A}$ of graphs that contain no odd hole, no antihole, and no "prism" (a graph consisting of two disjoint triangles with three disjoint paths between them). We prove that every graph $G\in{\cal A}$ different from a clique has an "even pair" (two vertices that are not joined by a chordless path of odd length), as conjectured by Everett and Reed [see the chapter "Even pairs" in the book {\it Perfect Graphs}, J.L. Ram\'{\i}rez-Alfons\'{\i}n and B.A. Reed, eds., Wiley Interscience, 2001]. Our proof is a polynomial-time algorithm that produces an even pair with the additional property that the contraction of this pair yields a graph in ${\cal A}$. This entails a polynomial-time algorithm, based on successively contracting even pairs, to color optimally every graph in ${\cal A}$. This generalizes several results concerning some classical families of perfect graphs.