Tournament immersion and cutwidth

Maria Chudnovsky, Alexandra Fradkin, Paul Seymour

Research output: Contribution to journalArticle

16 Scopus citations

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 {v1,...,vn} in such a way that for each j, there are at most k edges uv such that u∈{v1,...,vj-1} and v∈{vj,...,vn}.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.

Original languageEnglish (US)
Pages (from-to)93-101
Number of pages9
JournalJournal of Combinatorial Theory. Series B
Volume102
Issue number1
DOIs
StatePublished - Jan 1 2012

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Keywords

  • Cutwidth
  • Immersion
  • Tournament

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