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 work completes the structural description of $(2P_3,C_4,C_6)$-free graphs that contain no simplicial vertices by introducing and classifying the remaining challenging cases. It develops a framework based on $t$-pentagons and $t$-frames to show that such graphs either contain a simplicial or universal vertex or belong to one of a small family of basic graphs (5-baskets, villas, mansions, or 5-crowns). By applying a decomposition for Truemper configurations and leveraging bounded clique-width for these basic graphs, the authors derive a full description of all $(2P_3,C_4,C_6)$-free graphs with no simplicial vertices and prove bounded clique-width for the class, which in turn yields polynomial-time solvability of Graph Coloring for these graphs. The results culminate in a comprehensive classification: any graph in this family is obtainable from a base graph (thickening of a member of a fixed family or a basic graph) via universal vertex additions, ensuring tractable coloring and contributing to the broader understanding of even-hole-free and related graph classes.

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.

Figures (14)

• Figure 1.1: From left to right: graphs $4K_1$, $2P_3$, $C_4$, $C_6$, and $C_7$.
• Figure 1.2: Graphs $T_0$ (left) and $T_1$ (right).
• Figure 1.3: The $t$-pentagon for $t = 3$ (left), $t = 4$ (middle), and $t = 5$ (right).
• Figure 1.4: Three-path-configurations (3PCs): theta (left), pyramid (middle), and prism (right). A full line represents an edge, and a dashed line represents a path that has at least one edge.
• Figure 1.5: The only wheels in which all holes are of length five.

Theorems & Definitions (126)

• Theorem 3.1: Theorem 5.2 of 2P3C4C6FreePart1
• Theorem 3.2: Theorem 6.6 of 2P3C4C6FreePart1
• Proposition 4.1
• Proposition 4.2
• Proposition 4.3
• Proposition 4.4
• Proposition 4.5
• Proposition 4.6
• Proposition 4.7
• Proposition 4.8