Forbidden induced subgraphs for perfectness of claw-free graphs of independence number at least 4

TL;DR

This paper characterizes which graphs X make the class of connected, X-free, claw-free graphs with independence number at least 4 perfect, providing a complete classification and structural insights for certain cases.

Contribution

It provides a complete characterization of forbidden subgraphs X that ensure perfection in the specified graph class, including structural descriptions for particular cases.

Findings

All graphs are perfect iff X is an induced subgraph of P6, K1∪P5, 2P3, Z2, or K1∪Z1.

For X=2K1∪K3, all imperfect graphs are listed.

For other X, infinitely many imperfect graphs exist.

Abstract

For every graph $X$, we consider the class of all connected $\{K_{1,3}, X\}$-free graphs which are distinct from an odd cycle and have independence number at least $4$, and we show that all graphs in the class are perfect if and only if $X$ is an induced subgraph of some of $P_6$, $K_1 \cup P_5$, $2P_3$, $Z_2$ or $K_1 \cup Z_1$. Furthermore, for $X$ chosen as $2K_1 \cup K_3$, we list all eight imperfect graphs belonging to the class; and for every other choice of $X$, we show that there are infinitely many such graphs. In addition, for $X$ chosen as $B_{1,2}$, we describe the structure of all imperfect graphs in the class.