ModelingK-coteries by well-covered graphs

Yamashita, Masafumi; Kameda, Tiko

doi:10.1002/(sici)1097-0037(199910)34:3<221::aid-net7>3.0.co;2-m

Cited by 13 publications

(8 citation statements)

References 20 publications

(18 reference statements)

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“…Since every pendant edge is a strong clique, this implies that every well-covered graph of girth at least 8 is localizable. In 1999 [77], Yamashita and Kameda observed that all well-covered trees are localizable, pointed out that the converse inclusion fails in general, and asked for a characterization of localizable graphs.…”

Section: Introductionmentioning

confidence: 99%

Graphs vertex-partitionable into strong cliques

Hujdurović

Milanič

Ries

2018

Discrete Mathematics

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A graph is said to be well-covered if all its maximal independent sets are of the same size. In 1999, Yamashita and Kameda introduced a subclass of well-covered graphs, called localizable graphs and defined as graphs having a partition of the vertex set into strong cliques, where a clique in a graph is strong if it intersects all maximal independent sets. Yamashita and Kameda observed that all well-covered trees are localizable, pointed out that the converse inclusion fails in general, and asked for a characterization of localizable graphs.In this paper we obtain several structural and algorithmic results about localizable graphs. Our results include a proof of the fact that every very well-covered graph is localizable and characterizations of localizable graphs within the classes of line graphs, triangle-free graphs, C 4 -free graphs, and cubic graphs, each leading to a polynomial time recognition algorithm. On the negative side, we prove NP-hardness of recognizing localizable graphs within the classes of weakly chordal graphs, complements of line graphs, and graphs of independence number three. Furthermore, using localizable graphs we disprove a conjecture due to Zaare-Nahandi about kpartite well-covered graphs having all maximal cliques of size k. Our results unify and generalize several results from the literature.

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Section: Introductionmentioning

confidence: 99%

Graphs vertex-partitionable into strong cliques

Hujdurović

Milanič

Ries

2018

Discrete Mathematics

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“…Similarly, if a weight function w : V (G) −→ R is defined on the vertices of G, and G is w-well-covered, then finding a maximum weight independent set is a polynomial problem. There is an interesting application, where well-covered graphs are investigated in the context of distributed k-mutual exclusion algorithms [21].…”

Section: Well-covered Graphsmentioning

confidence: 99%

Generating subgraphs in chordal graphs

Levit¹,

Tankus²

2018

Preprint

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A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space, denoted W CW (G).Let B be a complete bipartite induced subgraph of G on vertex sets of bipartition B X and B Y . Then B is generating if there exists an independent set S such that S ∪ B X and S ∪ B Y are both maximal independent sets of G. In the restricted case that a generating subgraph B is isomorphic to K 1,1 , the unique edge in B is called a relating edge. Generating subgraphs play an important role in finding W CW (G).Deciding whether an input graph G is well-covered is co-NP-complete. Hence, finding W CW (G) is co-NP-hard. Deciding whether an edge is relating is NP-complete. Therefore, deciding whether a subgraph is generating is NP-complete as well.A graph is chordal if every induced cycle is a triangle. It is known that finding W CW (G) can be done polynomially in the restricted case that G is chordal. Thus recognizing well-covered chordal graphs is a polynomial problem. We present a polynomial algorithm for recognizing relating edges and generating subgraphs in chordal graphs.

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Section: Well-covered Graphsmentioning

confidence: 99%

Complexity Results for Generating Subgraphs

Levit

Tankus

2017

Algorithmica

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A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space, denoted W CW (G).Let B be a complete bipartite induced subgraph of G on vertex sets of bipartition BX and BY . Then B is generating if there exists an independent set S such that S∪BX and S∪BY are both maximal independent sets of G. In the restricted case that a generating subgraph B is isomorphic to K1,1, the unique edge in B is called a relating edge.Deciding whether an input graph G is well-covered is co-NP-complete. Therefore finding W CW (G) is co-NP-hard. Deciding whether an edge is relating is NP-complete. Therefore, deciding whether a subgraph is generating is NP-complete as well.In this article we discuss the connections among these problems, provide proofs for NP-completeness for several restricted cases, and present polynomial characterizations for some other cases.

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ModelingK-coteries by well-covered graphs

Cited by 13 publications

References 20 publications

Graphs vertex-partitionable into strong cliques

Graphs vertex-partitionable into strong cliques

Generating subgraphs in chordal graphs

Complexity Results for Generating Subgraphs

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