Degreewidth: A New Parameter for Solving Problems on Tournaments

Tom Davot, Lucas Isenmann, Sanjukta Roy, Jocelyn Thiebaut

Abstract: In the paper, we define a new parameter for tournaments called degreewidth which can be seen as a measure of how far is the tournament from being acyclic. The degreewidth of a tournament T denoted by \(\varDelta (T)\) is the minimum value k for which we can find an ordering \(\langle v_1, \dots , v_n \rangle \) of the vertices of T such that every vertex is incident to at most k backward arcs (i.e. an arc \((v_i,v_j)\) such that \(j<i\)). Thus, a tournament is acyclic if and only if its degreewidth is zero. Additionally, the class of sparse tournaments defined by Bessy et al. [ESA 2017] is exactly the class of tournaments with degreewidth one.