Product structure of graph classes with strongly sublinear separators

Abstract: We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more precise result, we show that if any hereditary graph class $\mathcal{G}$ admits $O(n<sup>{1-\epsilon})$ separators, then for any fixed $\delta\in(0,\epsilon)$ every $n$-vertex graph in $\mathcal{G}$ is a subgraph of the strong product of a graph $H$ with bounded tree-depth and a complete graph of size $O(n<sup>{1-\epsilon+\delta})$. This result holds with $\delta=0$ if we allow $H$ to have tree-depth $O(\log\log n)$. Moreover, using extensions of classical isoperimetric inequalties for grids graphs, we show the dependence on $\delta$ in our results and the above $\text{td}(H)\in O(\log\log n)$ bound are both best possible. We prove that $n$-vertex graphs of bounded treewidth are subgraphs of the product of a graph with tree-depth $t$ and a complete graph of size $O(n<sup>{1/t})$, which is best possible. Finally, we investigate the conjecture that for any hereditary graph class $\mathcal{G}$ that admits $O(n<sup>{1-\epsilon})$ separators, every $n$-vertex graph in $\mathcal{G}$ is a subgraph of the strong product of a graph $H$ with bounded tree-width and a complete graph of size $O(n<sup>{1-\epsilon})$. We prove this for various classes $\mathcal{G}$ of interest.