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Finding good 2-partitions of digraphs I. Hereditary properties

J Bang-Jensen 1 Frédéric Havet 2
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : We study the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties. Let H and E denote following two sets of natural properties of digraphs: H ={acyclic, complete, arcless, oriented (no 2-cycle), semicomplete, symmetric, tournament} and E ={strongly connected, connected, minimum out-degree at least 1, minimum in-degree at least 1, minimum semi-degree at least 1, minimum degree at least 1, having an out-branching, having an in-branching}. In this paper, we determine the complexity of of deciding, for any fixed pair of positive integers k1, k2, whether a given digraph has a vertex partition into two digraphs D1, D2 such that |V (Di)| ≥ ki and Di has property Pi for i = 1, 2 when P1 ∈ H and P2 ∈ H ∪ E. We also classify the complexity of the same problems when restricted to strongly connected digraphs. The complexity of the problems when both P1 and P2 are in E is determined in the companion paper [2].
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Submitted on : Monday, June 6, 2016 - 11:55:45 AM
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J Bang-Jensen, Frédéric Havet. Finding good 2-partitions of digraphs I. Hereditary properties. Theoretical Computer Science, Elsevier, 2016, 636, pp.85-94. ⟨10.1016/j.tcs.2016.05.029⟩. ⟨hal-01327015⟩

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