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Tournament pathwidth and topological containment. (English) Zbl 1301.05148
Summary: We prove that if a tournament has pathwidth $$\geq 4\theta^2+7\theta$$ then it has $$\theta$$ vertices that are pairwise $$\theta$$-connected. As a consequence of this and previous results, we obtain that for every set $$S$$ of tournaments the following are equivalent: (1) there exists $$k$$ such that every member of $$S$$ has pathwidth at most $$k$$, (2) there is a digraph $$H$$ such that no subdivision of $$H$$ is a subdigraph of any member of $$S$$, (3) there exists $$k$$ such that for each $$T\in S$$, there do not exist $$k$$ vertices of $$T$$ that are pairwise $$k$$-connected. As a further consequence, we obtain a polynomial-time algorithm to test whether a tournament contains a subdivision of a fixed digraph $$H$$ as a subdigraph.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments
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##### References:
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