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.
Categories: math.CO