Intersections of graphs and $\chi$-boundedness

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 $\chi$-boundedness. We consider the following two questions: Which graph classes have the property that their graph-intersection with every $\chi$-bounded class of graphs is $\chi$-bounded? We call such a class intersectionwise $\chi$-guarding. We prove that classes of graphs which admit a certain kind of decomposition are intersectionwise $\chi$-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 $\chi$-guarding, and we characterize the intersectionwise $\chi$-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 $\chi$-bounded? We call such a class intersectionwise self-$\chi$-guarding. We study intersectionwise self-$\chi$-guarding classes which are defined by a single forbidden induced subgraph, and we prove a result which allows us construct intersectionwise self-$\chi$-guarding classes from known intersectionwise $\chi$-guarding classes.