On the (Parameterized) Complexity of Recognizing Well-Covered \((r,\ell )\)-graphs

  • Sancrey Rodrigues Alves
  • Konrad K. Dabrowski
  • Luerbio Faria
  • Sulamita Klein
  • Ignasi Sau
  • Uéverton dos Santos Souza
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10043)


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


Well-covered graph \((r, \ell )\)-graph coNP-completeness FPT-algorithm Parameterized complexity Polynomial kernel 



We would like to thank the anonymous reviewers for their thorough, pertinent, and very helpful remarks.


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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Sancrey Rodrigues Alves
    • 1
  • Konrad K. Dabrowski
    • 2
  • Luerbio Faria
    • 3
  • Sulamita Klein
    • 4
  • Ignasi Sau
    • 5
    • 6
  • Uéverton dos Santos Souza
    • 7
  1. 1.FAETEC, Fundação de Apoio à Escola Técn. do Estado do Rio de JaneiroRio de JaneiroBrazil
  2. 2.School of Engineering and Computing SciencesDurham UniversityDurhamUK
  3. 3.UERJ, DICCUniversidade do Estado do Rio de JaneiroRio de JaneiroBrazil
  4. 4.UFRJ, COPPE-SistemasUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  5. 5.CNRS, LIRMMUniversité de MontpellierMontpellierFrance
  6. 6.Departamento de MatemáticaUniversidade Federal do CearáFortalezaBrazil
  7. 7.UFF, ICUniversidade Federal FluminenseNiteróiBrazil

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