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Other research product . 2022

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

Brause, Christoph; Doan, Trung Duy; Holub, Přemysl; Kabela, Adam; Ryjáček, Zdeněk; Schiermeyer, Ingo; Vrána, Petr;
Open Access
English
Published: 01 Jan 2022
Publisher: Elsevier
Country: Czech Republic
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 P6, K1 ∪ P5, 2P3, Z2 or K1 ∪ Z1. Furthermore, for X chosen as 2K1 ∪ K3, 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.

Subjects

perfect graphs, vertex colouring, forbidden induced subgraphs

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