Out-degree reducing partitions of digraphs

Joergen Bang-Jensen 1 Stéphane Bessy 2 Frédéric Havet 3 Anders Yeo 1
2 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
3 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : Let k be a fixed integer. We determine the complexity of finding a p-partition (V1,. .. , Vp) of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by Vi, (1 ≤ i ≤ p) is at least k smaller than the maximum out-degree of D. We show that this problem is polynomial-time solvable when p ≥ 2k and N P-complete otherwise. The result for k = 1 and p = 2 answers a question posed in [3]. We also determine, for all fixed non-negative integers k1, k2, p, the complexity of deciding whether a given digraph of maximum out-degree p has a 2-partition (V1, V2) such that the digraph induced by Vi has maximum out-degree at most ki for i ∈ [2]. It follows from this characterization that the problem of deciding whether a digraph has a 2-partition (V1, V2) such that each vertex v ∈ Vi has at least as many neighbours in the set V3−i as in Vi, for i = 1, 2 is N P-complete. This solves a problem from [6] on majority colourings.
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Joergen Bang-Jensen, Stéphane Bessy, Frédéric Havet, Anders Yeo. Out-degree reducing partitions of digraphs. Theoretical Computer Science, Elsevier, 2018, 719, pp.64-72. ⟨10.1016/j.tcs.2017.11.007⟩. ⟨hal-01765642⟩

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