Intersections of graphs and $χ$-boundedness

Abstract

Given $k$ graphs $G_{1}, \ldots, G_{k}$, their intersection is the graph $(\cap_{i\in [k]}V(G_{i}), \cap_{i\in [k]}E(G_{i}))$. Given $k$ graph classes $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$, we call the class $\{G: \forall i \in[k], \exists G_{i} \in \mathcal{G}_{i} \text{ such that } G=G_{1}\cap \ldots \cap G_{k}\}$ the graph-intersection of $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$. The main motivation for the work presented in this paper is to try to understand under which conditions graph-intersection preserves $χ$-boundedness. We consider the following two questions: Which graph classes have the property that their graph-intersection with every $χ$-bounded class of graphs is $χ$-bounded? We call such a class intersectionwise $χ$-guarding. We prove that classes of graphs which admit a certain kind of decomposition are intersectionwise $χ$-guarding. We provide necessary conditions that a finite set of graphs $\mathcal{H}$ should satisfy if the class of $\mathcal{H}$-free graphs is intersectionwise $χ$-guarding, and we characterize the intersectionwise $χ$-guarding classes which are defined by a single forbidden induced subgraph. Which graph classes have the property that, for every positive integer $k$, their $k$-fold graph-intersection is $χ$-bounded? We call such a class intersectionwise self-$χ$-guarding. We study intersectionwise self-$χ$-guarding classes which are defined by a single forbidden induced subgraph, and we prove a result which allows us construct intersectionwise self-$χ$-guarding classes from known intersectionwise $χ$-guarding classes.