The class of $(2P_3,C_4,C_6)$-free graphs, part II: $(2P_3,C_4,C_6,C_7,T_0)$-free graphs

TL;DR

This paper fully characterizes a class of graphs defined by forbidden induced subgraphs, proving they have bounded clique-width, which leads to polynomial-time algorithms for graph coloring within this class.

Contribution

It provides a complete structural description of $(2P_3,C_4,C_6)$-free graphs without simplicial vertices and proves they have bounded clique-width, enabling efficient coloring algorithms.

Findings

Graphs in the class have bounded clique-width.

Graph coloring can be solved in polynomial time for these graphs.

Structural characterization aids in understanding graph properties.

Abstract

This is the second in a series of two papers dealing with $(2P_3,C_4,C_6)$-free graphs, or equivalently, $(2P_3,\text{even hole})$-free graphs. In this two-paper series, we give a full structural description of $(2P_3,C_4,C_6)$-free graphs that contain no simplicial vertices, and we show that such graphs have bounded clique-width. This implies that Graph Coloring can be solved in polynomial time for $(2P_3,C_4,C_6)$-free graphs. In the first paper of the series, we described the structure of $(2P_3,C_4,C_6)$-free graphs that contain an induced $C_7$ or an induced $T_0$ (where $T_0$ is a certain 2-connected graph on nine vertices in which all holes are of length five), and we showed that such graphs either contain a simplicial vertex or have bounded clique-width. In the present paper (the second part of the series), we describe the structure of $(2P_3,C_4,C_6,C_7,T_0)$-free graphs that contain no simplicial vertices, and we show that such graphs have bounded clique-width. Finally this paper gives the full statement of the theorem describing the structure of $(2P_3,C_4,C_6)$-free graphs that contain no simplicial vertices.