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Vertex Partitions of Graphs into Cographs and Stars

Paul Dorbec 1 Mickael Montassier 2 Pascal Ochem 3
1 Combinatoire et Algorithmique
LaBRI - Laboratoire Bordelais de Recherche en Informatique
3 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : A cograph is a graph that contains no path on four vertices as an induced subgraph. A cograph k-partition of a graph G = (V, E) is a vertex-partition of G into k sets V1 , . . . , Vk ⊂ V so that the graph induced by Vi is a cograph for 1 ≤ i ≤ k. Gimbel and Nešetril [5] studied the complexity aspects of the cograph k-partitions and raised the following questions: Does there exist a triangle-free planar graph that is not cograph 2-partitionable? If the answer is yes, what is the complexity of the associated decision problem? In this paper, we prove that such an example exists and that deciding whether a triangle-free planar graph admits a cograph 2-partition is NP-complete. We also show that every graph with maximum average degree at most 14/5 admits a cograph 2-partition such that each component is a star on at most three vertices.
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Submitted on : Friday, November 29, 2013 - 9:51:05 AM
Last modification on : Thursday, November 8, 2018 - 3:26:18 PM

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Paul Dorbec, Mickael Montassier, Pascal Ochem. Vertex Partitions of Graphs into Cographs and Stars. Journal of Graph Theory, Wiley, 2013, 75, pp.75-90. ⟨10.1002/jgt.21724⟩. ⟨hal-00911272⟩



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