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

Alves, Sancrey Rodrigues; Dabrowski, Konrad K.; Faria, Luerbio; Klein, Sulamita; Sau, Ignasi; Souza, Uéverton S.

Abstract

An (r,ℓ)(r,ℓ)-partition of a graph G is a partition of its vertex set into r independent sets and ℓ 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)wc-g 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, and the question is whether G is well-covered. This generates two infinite families of problems, for any fixed non-negative integers r and ℓ, which we classify as being P, coNP-complete, NP-complete, NP-hard, or coNP-hard. Only the cases wc-(r,0)(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 α of a maximum independent set of the input graph, its neighborhood diversity, its clique-width, or the number ℓ of cliques in an (r,ℓ)(r,ℓ)-partition. In particular, we show that the parameterized problem of determining whether every maximal independent set of an input graph G has cardinality equal to k can be reduced to the wc-(0,ℓ)(0,ℓ)g problem parameterized by ℓ. In addition, we prove that both problems are coW[2]-hard but can be solved in XP-time.

Citation

Alves, S. R., Dabrowski, K. K., Faria, L., Klein, S., Sau, I., & Souza, U. S. (2018). On the (Parameterized) Complexity of Recognizing Well-covered ( r , ℓ )-graph. Theoretical Computer Science, 746, 36-48. https://doi.org/10.1016/j.tcs.2018.06.024

• Journal Article Type: Article
• Acceptance Date: Jun 12, 2018
• Online Publication Date: Jun 22, 2018
• Publication Date: Oct 25, 2018
• Deposit Date: Jun 26, 2018
• Publicly Available Date: Jun 22, 2019
• Journal: Theoretical Computer Science
• Print ISSN: 0304-3975
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 746
• Pages: 36-48
• DOI: https://doi.org/10.1016/j.tcs.2018.06.024
• Public URL: https://durham-repository.worktribe.com/output/1323527