On the structure of ($4K_1$, $C_4$, $P_6$)-free graphs

TL;DR

This paper proves that ($4K_1$, $C_4$, $P_6$)-free graphs containing a $C_6$ have bounded clique-width, leading to a polynomial-time coloring algorithm for these graphs, advancing understanding of their structural complexity.

Contribution

The paper introduces a new method to bound clique-width in ($4K_1$, $C_4$, $P_6$)-free graphs and establishes polynomial-time coloring algorithms for them.

Findings

($4K_1$, $C_4$, $P_6$)-free graphs with a $C_6$ have bounded clique-width

Bounded clique-width implies polynomial-time coloring algorithms

New method for bounding clique-width of complex graph classes

Abstract

Determining the complexity of colouring ($4K_1, C_4$)-free graph is a long open problem. Recently Penev showed that there is a polynomial-time algorithm to colour a ($4K_1, C_4, C_6$)-free graph. In this paper, we will prove that if $G$ is a ($4K_1, C_4, P_6$)-free graph that contains a $C_6$, then $G$ has bounded clique-width. To this purpose, we use a new method to bound the clique-width, that is of independent interest. As a consequence, there is a polynomial-time algorithm to colour ($4K_1, C_4, P_6$)-free graphs.