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

TL;DR

The paper tackles the coloring problem for ($4K_1$, $C_4$)-free graphs by studying the class ($4K_1$, $C_4$, $P_6$)-free graphs. It develops a detailed structural analysis around an induced $C_6$ to decompose the graph into well-behaved parts and introduces a novel bounding technique based on monotone partitions with the hev-property, proving bounded clique-width (≤27) for graphs containing a $C_6$. This structural bound, together with Rao’s framework, yields a polynomial-time coloring algorithm for ($4K_1$, $C_4$, $P_6$)-free graphs. The work also connects to and extends prior results on related forbidden-vertex classes and suggests directions for generalizing the hev-property approach to broader graph families.

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.

Figures (2)

• Figure 1: All four-vertex graphs
• Figure 2: Adjacencies in $X_3\cup X_4$. Solid line (resp. dashed line) between two sets $A$ and $B$ means that $A$ is complete (resp. monotone) to $B$.

Theorems & Definitions (14)

• Theorem 1.1: Pen2020
• Theorem 1.2
• Theorem 2.1: Rao
• Theorem 2.2: Pen2020
• Theorem 2.3: FolFra2020
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Lemma 2.7
• Lemma 2.8