Very well covered graphs

Favaron, Odile

doi:10.1016/0012-365x(82)90215-1

Cited by 79 publications

(63 citation statements)

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“…Such a set can be obtained in polynomial time and, hence, the dominating set problem is in P for the family W AR . We acknowledge that our approach in solving the above problem is based on the one that Favaron [4] used to solve the same problem for very well-covered graphs. G H has a Hamiltonian path.…”

Section: Its Neighbor Set Is Made Up Of Only Those Vertices Thatmentioning

confidence: 99%

“…ognition problem is in P. These include very well covered graphs [4], well-covered line graphs [7], well-covered cubic graphs [2], well-covered graphs with girth ¢ 5 As in the case of the family W SR , all recursive decom- [6], well-covered graphs with no 4-or 5-cycles [5], wellpositions of a graph G √ W AR yield the same set of layers covered perfect graphs with bounded clique size and welland these layers satisfy properties (a), (b), and (c) of covered perfect graphs with no induced 4-cycles [3], Theorem 2.5 (and also of Theorem 2.3). We define a set claw-free well-covered graphs with no 4-cycles [15], and of subclasses of W AR as follows:…”

Section: And the Parts Of L Jmentioning

confidence: 99%

“…tion, and dominating set problems for very well covered Plummer [8] called a graph well covered if every maxgraphs follows from Favaron's results on this family [4]. imal independent set in it has the same size.…”

Section: Introductionmentioning

confidence: 97%

“…In this paper, we examine two subclasses of well-covered [14], Ravindra [10], and Favaron [4] studied this family.…”

Section: Introductionmentioning

confidence: 99%

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Tracking the P-NP boundary for well-covered graphs

Sankaranarayana

1997

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Section: Its Neighbor Set Is Made Up Of Only Those Vertices Thatmentioning

confidence: 99%

Section: And the Parts Of L Jmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

“…In this paper, we examine two subclasses of well-covered [14], Ravindra [10], and Favaron [4] studied this family.…”

Section: Introductionmentioning

confidence: 99%

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Tracking the P-NP boundary for well-covered graphs

Sankaranarayana

1997

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“…Note that the only remaining open cases are wc(r, 0)g for r ≥ 3. As an avenue for further research, it would be interesting to provide a complete characterization of well-covered tripartite graphs, as has been done for bipartite graphs [10,21,26]. So far, only partial characterizations exist [14,15].…”

Section: Factmentioning

confidence: 99%

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

Alves

Dabrowski

Faria

et al. 2016

Combinatorial Optimization and Applications

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Citation for published item:elvesD nrey odrigues nd hrowskiD uonrd uF nd priD vuerio nd uleinD ulmit nd uD sgnsi nd dos ntos ouzD ¡ everton @PHITA 9yn the @prmeterizedA omplexity of reognizing wellEovered @rDlAEgrphsF9D in gomintoril optimiztion nd pplitions X IHth snterntionl gonfereneD gygye PHITD rong uongD ghinD heemer IT!IVD PHIT Y proeedingsF ghmD witzerlndX pringerD ppF RPQERQUF veture notes in omputer sieneF @IHHRQAF Further information on publisher's website:Publisher's copyright statement:The nal publication is available at Springer via https://doi.org/10.1007/978-3-319-48749-631Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. An (r, )-partition of a graph G is a partition of its vertex set into r independent sets and cliques. A graph is (r, ) if it admits an (r, )-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is (r, )-well-covered if it is both (r, ) and well-covered. In this paper we consider two different decision problems. In the (r, )-Well-Covered Graph problem ((r, )wcg for short), we are given a graph G, and the question is whether G is an (r, )-well-covered graph. In the Well-Covered (r, )-Graph problem (wc(r, )g for short), we are given an (r, )-graph G together with an (r, )-partition of V (G) into r independent sets and 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 ≥ 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, or the number of cliques in an (r, )-partition. In particular, we show that the parameterized problem of deciding whether a general graph is well-covered parameterized by α can be reduced to the wc(0, )g problem parameterized by , and we prove that this latter problem is in XP but does not admit polynomial kernels unless coNP ⊆ NP/poly.

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ModelingK-coteries by well-covered graphs

Yamashita

Kameda

1999

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Very well covered graphs

Cited by 79 publications

References 2 publications

Tracking the P-NP boundary for well-covered graphs

Tracking the P-NP boundary for well-covered graphs

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

ModelingK-coteries by well-covered graphs

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