Tournament pathwidth and topological containment

We prove that for every set S of tournaments the following are equivalent: • there exists k such that every member of S has pathwidth at most k • there is a digraph H such that no subdivision of H is a subdigraph of any member of S • there exists k such that for each T ∈ S, there do not exist k vertices of T that are pairwise k-connected. As a consequence, we obtain a polynomial-time algorithm to test whether a tournament contains a subdivision of a fixed digraph H as a subdigraph.