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On the (Parameterized) Complexity of Recognizing Well-covered (r,l)-graphs

Alves, Sancrey Rodrigues; Dabrowski, Konrad K.; Faria, Luerbio; Klein, Sulamita; Sau, Ignasi; dos Santos Souza, Uéverton; Chan, T.-H. Hubert; Li, Minming; Wang, Lusheng

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Sancrey Rodrigues Alves

Konrad K. Dabrowski

Luerbio Faria

Sulamita Klein

Ignasi Sau

Uéverton dos Santos Souza

T.-H. Hubert Chan

Minming Li

Lusheng Wang


An (r,ℓ)(r,ℓ)-partition of a graph G is a partition of its vertex set into r independent sets and ℓℓ cliques. A graph is (r,ℓ)(r,ℓ) if it admits an (r,ℓ)(r,ℓ)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is (r,ℓ)(r,ℓ)-well-covered if it is both (r,ℓ)(r,ℓ) and well-covered. In this paper we consider two different decision problems. In the (r,ℓ)(r,ℓ)-Well-Covered Graph problem ((r,ℓ)(r,ℓ) wcg for short), we are given a graph G, and the question is whether G is an (r,ℓ)(r,ℓ)-well-covered graph. In the Well-Covered (r,ℓ)(r,ℓ)-Graph problem (wc (r,ℓ)(r,ℓ) g for short), we are given an (r,ℓ)(r,ℓ)-graph G together with an (r,ℓ)(r,ℓ)-partition of V(G) into r independent sets and ℓℓ cliques, and the question is whether G is well-covered. We classify most of these problems into P, coNP-complete, NP-complete, NP-hard, or coNP-hard. Only the cases wc(r, 0)g for r≥3r≥3 remain open. In addition, we consider the parameterized complexity of these problems for several choices of parameters, such as the size αα of a maximum independent set of the input graph, its neighborhood diversity, or the number ℓℓ of cliques in an (r,ℓ)(r,ℓ)-partition. In particular, we show that the parameterized problem of deciding whether a general graph is well-covered parameterized by αα can be reduced to the wc (0,ℓ)(0,ℓ) g problem parameterized by ℓℓ, and we prove that this latter problem is in XP but does not admit polynomial kernels unless coNP⊆NP/polycoNP⊆NP/poly.

Presentation Conference Type Conference Paper (Published)
Conference Name 10th Annual International Conference on Combinatorial Optimization and Applications (COCOA 2016)
Start Date Dec 16, 2016
End Date Dec 18, 2016
Acceptance Date Sep 2, 2016
Online Publication Date Oct 31, 2016
Publication Date Oct 31, 2016
Deposit Date Nov 26, 2016
Publicly Available Date Oct 31, 2017
Pages 423-437
Series Title Lecture notes in computer science
Series Number 10043
Series ISSN 0302-9743,1611-3349
Book Title Combinatorial optimization and applications : 10th International Conference, COCOA 2016, Hong Kong, China, December 16–18, 2016 ; proceedings.
ISBN 9783319487489
Public URL


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