Strongly sublinear separators and bounded asymptotic dimension for sphere intersection graphs

Abstract: In this paper, we consider the class $\mathcal{C}<sup>d$ of sphere intersection graphs in $\mathbb{R}<sup>d$ for $d \geq 2$. We show that for each integer $t$, the class of all graphs in $\mathcal{C}<sup>d$ that exclude $K_{t,t}$ as a subgraph has strongly sublinear separators. We also prove that $\mathcal{C}<sup>d$ has asymptotic dimension at most $2d+2$.