Making a Tournament Indecomposable by One Subtournament-Reversal Operation

Abstract

Given a tournament T, a module of T is a subset M of V(T) such that for \(x, y\in M\) and \(v\in V(T)\setminus M\), \((v,x)\in A(T)\) if and only if \((v,y)\in A(T)\). The trivial modules of T are \(\varnothing\), \(\{u\}\) \((u\in V(T))\) and V(T). The tournament T is indecomposable if all its modules are trivial; otherwise it is decomposable. Let T be a tournament with at least five vertices. In a previous paper, the authors proved that the smallest number \(\delta (T)\) of arcs that must be reversed to make T indecomposable satisfies \(\delta (T) \le \left\lceil \frac{v(T)+1}{4} \right\rceil\), and this bound is sharp, where \(v(T) = |V(T)|\) is the order of T. In this paper, we prove that if the tournament T is not transitive of even order, then T can be made indecomposable by reversing the arcs of a subtournament of T. We denote by \(\delta '(T)\) the smallest size of such a subtournament. We also prove that \(\delta (T) = \left\lceil \frac{\delta '(T)}{2} \right\rceil\).

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Correspondence to Houmem Belkhechine.

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Belkhechine, H., Ben Salha, C. Making a Tournament Indecomposable by One Subtournament-Reversal Operation. Graphs and Combinatorics (2021). https://doi.org/10.1007/s00373-021-02282-0

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Keywords

  • Module
  • Co-module
  • Indecomposable
  • Decomposability arc-index
  • Decomposability subtournament-index
  • Co-modular index

Mathematics Subject Classification

  • 05C20
  • 05C35