On the structure of (banner, odd hole)-free graphs

TL;DR

This paper characterizes the structure of (banner, odd hole)-free graphs, proves they are perfect-divisible, and provides polynomial-time algorithms for recognition, coloring, and finding maximum stable sets in such graphs.

Contribution

It introduces a structural characterization of (banner, odd hole)-free graphs and develops efficient algorithms for recognition, coloring, and stable set problems.

Findings

Such graphs are either perfect, contain no stable set of size three, or have a homogeneous set.

They are proven to be perfect-divisible, bounding their chromatic number.

Polynomial-time algorithms are provided for recognition, coloring, and maximum stable set detection.

Abstract

A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A banner is a graph which consists of a hole on four vertices and a single vertex with precisely one neighbor on the hole. We prove that a (banner, odd hole)-free graph is perfect, or does not contain a stable set on three vertices, or contains a homogeneous set. Using this structure result, we design a polynomial-time algorithm for recognizing (banner, odd hole)-free graphs. We also design polynomial-time algorithms to find, for such a graph, a minimum coloring and largest stable set. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$ is bounded by $\omega^2$ where $\omega$ denotes the number of vertices in a largest clique of $G$. We prove that (banner, odd hole)-free graphs are perfect-divisible. %