# Degreewidth: a New Parameter for Solving Problems on Tournaments

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 Δ(T) is the minimum value k for which we can find an ordering ⟨ v_1, …, v_n ⟩ 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. We first study computational complexity of finding degreewidth. Namely, we show it is NP-hard and complement this result with a 3-approximation algorithm. We also provide a cubic algorithm to decide if a tournament is sparse. Finally, we study classical graph problems Dominating Set and Feedback Vertex Set parameterized by degreewidth. We show the former is fixed parameter tractable whereas the latter is NP-hard on sparse tournaments. Additionally, we study Feedback Arc Set on sparse tournaments.

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