Edge separators for graphs excluding a minor

TL;DR

This paper establishes a bound on the size of edge separators in $K_t$-minor-free graphs with bounded degree, extending previous planar and bounded-genus results to a broader class of graphs.

Contribution

It introduces a new edge separator bound for $K_t$-minor-free graphs with maximum degree, generalizing earlier planar and genus-specific results.

Findings

Edge separator size is $O(t^2(\log t)^{1/4}\sqrt{\Delta n})$.

Line graph of $G$ embeds into a strong product with bounded treewidth.

Results are tight up to the dependency on $t$.

Abstract

We prove that every $n$-vertex $K_t$-minor-free graph $G$ of maximum degree $\Delta$ has a set $F$ of $O(t^2(\log t)^{1/4}\sqrt{\Delta n})$ edges such that every component of $G - F$ has at most $n/2$ vertices. This is best possible up to the dependency on $t$ and extends earlier results of Diks, Djidjev, Sykora, and Vr\v{t}o (1993) for planar graphs, and of Sykora and Vr\v{t}o (1993) for bounded-genus graphs. Our result is a consequence of the following more general result: The line graph of $G$ is isomorphic to a subgraph of the strong product $H \boxtimes K_{\lfloor p \rfloor}$ for some graph $H$ with treewidth at most $t-2$ and $p = \sqrt{(t-3)\Delta |E(G)|} + \Delta$.