Structure and algorithms for (cap, even hole)-free graphs

TL;DR

This paper studies (cap, even hole)-free graphs, providing structural insights, bounds on chromatic number, and efficient algorithms for coloring and maximum weight stable set problems.

Contribution

It offers an explicit construction, proves degree and chromatic bounds, and develops polynomial algorithms for coloring and stable set problems in these graph classes.

Findings

Every such graph has a vertex of degree at most (3/2) times the clique number minus one.

Provides an O(nm) algorithm for q-coloring fixed q.

Establishes bounds on treewidth and clique-width for these graphs.

Abstract

A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph $G$ has a vertex of degree at most $\frac{3}{2}\omega (G) -1$, and hence $\chi(G)\leq \frac{3}{2}\omega (G)$, where $\omega(G)$ denotes the size of a largest clique in $G$ and $\chi(G)$ denotes the chromatic number of $G$. We give an $O(nm)$ algorithm for $q$-coloring these graphs for fixed $q$ and an $O(nm)$ algorithm for maximum weight stable set. We also give a polynomial-time algorithm for minimum coloring. Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs $G$ without clique cutsets have treewidth at most $6\omega(G)-1$ and clique-width at most 48.