2015
DOI: 10.1016/j.dam.2015.01.001
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Well-covered graphs without cycles of lengths 4, 5 and 6

Abstract: Let G be a graph. A set S of vertices in G dominates the graph if every vertex of G is either in S or a neighbor of a vertex in S. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph G is well-dominated if all its minimal dominating sets are of the same cardinality. The complexity status of recognizing well-dominated graphs is not known. We show that recognizing well-dominated graphs can be done polynomially for graphs without cycles of lengths 4 and 5, by proving t… Show more

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Cited by 16 publications
(26 citation statements)
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“…It is proved in [7] and [9] that recognizing generating subgraphs can be done polynomially for G( C 4 , C 6 , C 7 ), and G( C 5 , C 6 , C 7 ). Theorem 1.3 [7] The following problem can be solved in polynomial time:…”
Section: Question: Is Xy a Relating Edge?mentioning
confidence: 99%
See 2 more Smart Citations
“…It is proved in [7] and [9] that recognizing generating subgraphs can be done polynomially for G( C 4 , C 6 , C 7 ), and G( C 5 , C 6 , C 7 ). Theorem 1.3 [7] The following problem can be solved in polynomial time:…”
Section: Question: Is Xy a Relating Edge?mentioning
confidence: 99%
“…Question: Is B generating? Theorem 1.4 [9] The following problem can be solved in polynomial time: A relating edge is a restricted case of a genereting subgraph. However, the complexity of the algorithm for recognizing related edges, presented in Section 3, is O (|V | (|V | + |E|)), while the complexity of the algorithm which recognizes generating subgraphs in Section 4 is O |V | 2 (|V | + |E|) .…”
Section: Question: Is Xy a Relating Edge?mentioning
confidence: 99%
See 1 more Smart Citation
“…Since recognizing well-covered graphs is co-NP-complete, finding the vector space W CW (G) of an input graph G is co-NP-hard. In [14] there is a polynomial algorithm which receives as its input a graph G without cycles of lengths 4, 5, and 6, and outputs W CW (G).…”
Section: Well-covered Graphsmentioning
confidence: 99%
“…Finbow et al [2] showed that a bipartite well-dominated graph is either in the family P or is a 4-cycle. Further characterization results in the literature are for well-dominated block graphs and unicyclic graphs [9], locally well-dominated graphs and locally independent well-dominated graphs [10], 3-connected, planar, and claw-free well-dominated graphs [5], 4-connected, 4-regular, claw-free well-dominated graphs [3], and well-dominated graphs containing neither 4-cycles nor 5-cycles [6].…”
Section: Introductionmentioning
confidence: 99%