A characterization of b-perfect graphs

TL;DR

This paper characterizes b-perfect graphs by identifying a finite set of forbidden induced subgraphs, enabling polynomial-time recognition and coloring algorithms for these graphs.

Contribution

It provides a complete characterization of b-perfect graphs through forbidden subgraphs, along with efficient recognition and coloring algorithms.

Findings

Characterization of b-perfect graphs via 22 forbidden induced subgraphs

Existence of polynomial-time recognition algorithm for b-perfect graphs

Polynomial-time algorithm for optimal coloring of b-perfect graphs

Abstract

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number of a graph $G$ is the largest integer $k$ such that $G$ admits a b-coloring with $k$ colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph of $G$. We prove that a graph is b-perfect if and only if it does not contain as an induced subgraph a member of a certain list of twenty-two graphs. This entails the existence of a polynomial-time recognition algorithm and of a polynomial-time algorithm for coloring exactly the vertices of every b-perfect graph.