The class of $(2P_3,C_4,C_6)$-free graphs, part I: $(2P_3,C_4,C_6)$-free graphs that contain an induced $C_7$ or an induced $T_0$

TL;DR

This work develops a comprehensive structural description for the class of $(2P_3,C_4,C_6)$-free graphs that contain an induced $C_7$ or $T_0$, showing such graphs either possess a simplicial vertex or have bounded clique-width (specifically, at most 12 for the no-simplicial-vertices subfamily). The authors introduce the 7-saucer and tent configurations, plus a finite base family ${\mathcal M}$, and prove that graphs in the class decompose into either these canonical forms or thickened versions with optional universal vertices. This yields a pathway to polynomial-time Graph Coloring for the class via bounded clique-width, and an $O(n^3)$ recognition and coloring framework for the studied subfamilies. The results lay the groundwork for the second paper, which completes the description of all $(2P_3,C_4,C_6)$-free graphs that contain no simplicial vertices and establishes bounded clique-width for the remaining cases. Overall, the work advances the understanding of even-hole-free structures and provides practical algorithmic consequences for coloring in this graph family.

Abstract

This is the first 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 this paper, we describe 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 show that such graphs either contain a simplicial vertex or have bounded clique-width. In the second part of this series, we describe the structure of all $(2P_3,C_4,C_6,C_7,T_0)$-free graphs that contain no simplicial vertices, and we show that such graphs have bounded clique-width. The full statement of the theorem describing the structure of $(2P_3,C_4,C_6)$-free graphs that contain no simplicial vertices is given in the second paper of this series.

Figures (6)

• Figure 1.1: From left to right: graphs $4K_1$, $2P_3$, $C_4$, $C_6$, and $C_7$.
• Figure 1.2: The 3-pentagon (left), $T_0$ (center), and $T_1$ (right).
• Figure 1.3: 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 1.4: Two copies of the same $(P_7,C_4,C_6,C_7)$-free graph on eleven vertices, with induced 3-pentagons intersecting in an inconvenient way. Both full and dashed lines represent edges. The edges of two induced 3-pentagons are indicated by dashed lines.
• Figure 1.5: The only wheels in which all holes are of length five.

Theorems & Definitions (112)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof