Explicit bounds for graph minors

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.