Separator Theorems for Minor-Free and Shallow Minor-Free Graphs with Applications

Alon, Seymour, and Thomas generalized Lipton and Tarjan's planar separator theorem and showed that a $K_h$-minor free graph with $n$ vertices has a separator of size at most $h^{3/2}\sqrt n$. They gave an algorithm that, given a graph $G$ with $m$ edges and $n$ vertices and given an integer $h\geq 1$, outputs in $O(\sqrt{hn}m)$ time such a separator or a $K_h$-minor of $G$. Plot kin, Rao, and Smith gave an $O(hm\sqrt{n\log n})$ time algorithm to find a separator of size $O(h\sqrt{n\log n})$. Kawara bayashi and Reed improved the bound on the size of the separator to $h\sqrt n$ and gave an algorithm that finds such a separator in $O(n^{1 + \epsilon})$ time for any constant $\epsilon >, 0$, assuming $h$ is constant. This algorithm has an extremely large dependency on $h$ in the running time (some power tower of $h$ whose height is itself a function of $h$), making it impractical even for small $h$. We are interested in a small polynomial time dependency on $h$ and we show how to find an $O(h\sqrt{n\log n})$-size separator or report that $G$ has a $K_h$-minor in $O(\poly(h)n^{5/4 + \epsilon})$ time for any constant $\epsilon >, 0$. We also present the first $O(\poly(h)n)$ time algorithm to find a separator of size $O(n^c)$ for a constant $c