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.
We study computational complexity of finding degreewidth. We show it is NP-hard and complement this result with a 3-approximation algorithm. We provide a \(O(n^3)\)-time algorithm to decide if a tournament is sparse, where n is its number of vertices.
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 even on sparse tournaments. Additionally, we show polynomial time algorithm for Feedback Arc Set on sparse tournaments.
Sanjukta Roy was affiliated to Faculty of Information Technology, Czech Technical University in Prague when majority of this work was done. Jocelyn Thiebaut was supported by the CTU Global postdoc fellowship program.
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Notes
- 1.
Not to be confused with sparse tournaments that has an arc between every pair of vertices, hence, is not a sparse graph.
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Acknowledgements
We would like to thank Frédéric Havet for pointing us a counter-example to the polynomial running-time algorithm in [24, Lemma 35.1, p. 97].
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Davot, T., Isenmann, L., Roy, S., Thiebaut, J. (2023). Degreewidth: A New Parameter for Solving Problems on Tournaments. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_18
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