Substitution and $\chi$-Boundedness

TL;DR

This paper proves that certain graph classes remain $ ext{chi}$-bounded when closed under specific operations like substitution and gluing, and that polynomial or exponential bounds are preserved under substitution.

Contribution

It establishes that $ ext{chi}$-boundedness is preserved under substitution and gluing operations, extending known properties and bounds to these graph class closures.

Findings

Closure under substitution preserves $ ext{chi}$-boundedness with polynomial/exponential bounds.

Closure under gluing along a clique or bounded vertices preserves $ ext{chi}$-boundedness.

Polynomial/exponential bounds are maintained under substitution in $ ext{chi}$-bounded classes.

Abstract

A class $\mathcal{G}$ of graphs is said to be {\em $\chi$-bounded} if there is a function $f:\mathbb{N} \rightarrow \mathbb{R}$ such that for all $G \in \mathcal{G}$ and all induced subgraphs $H$ of $G$, $\chi(H) \leq f(\omega(H))$. In this paper, we show that if $\mathcal{G}$ is a $\chi$-bounded class, then so is the closure of $\mathcal{G}$ under any one of the following three operations: substitution, gluing along a clique, and gluing along a bounded number of vertices. Furthermore, if $\mathcal{G}$ is $\chi$-bounded by a polynomial (respectively: exponential) function, then the closure of $\mathcal{G}$ under substitution is also $\chi$-bounded by some polynomial (respectively: exponential) function. In addition, we show that if $\mathcal{G}$ is a $\chi$-bounded class, then the closure of $\mathcal{G}$ under the operations of gluing along a clique and gluing along a bounded number of vertices together is also $\chi$-bounded, as is the closure of $\mathcal{G}$ under the operations of substitution and gluing along a clique together.