# Tournament immersion and cutwidth

## Author(s): Chudnovsky, Maria; Fradkin, Alexandra; Seymour, Paul D.

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 Abstract: A (loopless) digraph H is strongly immersed in a digraph G if the vertices of H are mapped to distinct vertices of G, and the edges of H are mapped to directed paths joining the corresponding pairs of vertices of G, in such a way that the paths used are pairwise edge-disjoint, and do not pass through vertices of G that are images of vertices of H. A digraph has cutwidth at most k if its vertices can be ordered (v(1), ... , v(n)) in such a way that for each j, there are at most k edges uv such that u is an element of v(1), ... , v(j-1) and v is an element of v(j), ... , v(n). We prove that for every set S of tournaments, the following are equivalent: there is a digraph H such that H cannot be strongly immersed in any member of S. there exists k such that every member of S has cutwidth at most k, there exists k such that every vertex of every member of S belongs to at most k edge-disjoint directed cycles. This is a key lemma towards two results that will be presented in later papers: first, that strong immersion is a well-quasi-order for tournaments, and second, that there is a polynomial time algorithm for the k edge-disjoint directed paths problem (for fixed k) in a tournament. (C) 2011 Elsevier Inc. All rights reserved. Publication Date: Jan-2012 Electronic Publication Date: 1-Jun-2011 Citation: Chudnovsky, Maria, Fradkin, Alexandra, Seymour, Paul. (2012). Tournament immersion and cutwidth. JOURNAL OF COMBINATORIAL THEORY SERIES B, 102 (93 - 101. doi:10.1016/j.jctb.2011.05.001 DOI: doi:10.1016/j.jctb.2011.05.001 ISSN: 0095-8956 Pages: 93 - 101 Type of Material: Journal Article Journal/Proceeding Title: JOURNAL OF COMBINATORIAL THEORY SERIES B Version: Final published version. Article is made available in OAR by the publisher's permission or policy.

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