Explicit bounds for graph minors

TL;DR

This paper establishes explicit bounds on how certain graph structures can be manipulated on surfaces, leading to concrete constants in graph minor algorithms and results on redundant vertices.

Contribution

It provides explicit bounds and constants in graph minor theory on surfaces, improving previous qualitative results with computable, quantitative measures.

Findings

Proves a homeomorphism with bounded intersection properties for surface paths.

Derives explicit constants in Robertson and Seymour's graph minor algorithms.

Shows that t-protected vertices are redundant with explicit bounds.

Abstract

Let $\Sigma$ be a surface with boundary $b(\Sigma)$, $\mathcal{L}$ be a collection of $k$ disjoint $b(\Sigma)$-paths in $\Sigma$, and $P$ be a non-separating $b(\Sigma)$-path in $\Sigma$. We prove that there is a homeomorphism $\phi: \Sigma \to \Sigma$ that fixes each point of $b(\Sigma)$ and such that $\phi(\mathcal{L})$ meets $P$ at most $2k$ times. With this theorem, we derive explicit constants in the graph minor algorithms of Robertson and Seymour. We reprove a result concerning redundant vertices for graphs on surfaces, but with explicit bounds. That is, we prove that there exists a computable integer $t:=t(\Sigma,k)$ such that if $v$ is a '$t$-protected' vertex in a surface $\Sigma$, then $v$ is redundant with respect to any $k$-linkage.