The Well-Covered Dimension of Products of Graphs

Birnbaum, Isaac; Kuneli, Megan; McDonald, Robyn; Urabe, Katherine; Vega, Óscar Gerardo Alvarado

doi:10.48550/arxiv.1003.3968

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2015

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“…The dimension of W CW (G) is denoted by wcdim(G) [3]. More recent results about wcdim can be found in [1] and [2].…”

Section: Andmentioning

confidence: 99%

Well-covered graphs without cycles of lengths 4, 5 and 6

Levit

Tankus

2015

Discrete Applied Mathematics

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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 that a graph belonging to this family is well-dominated if and only if it is well-covered.Assume that a weight function w is defined on the vertices of G. Then G is w-well-dominated if all its minimal dominating sets are of the same weight. We prove that the set of weight functions w such that G is w-welldominated is a vector space, and denote that vector space by W W D(G). We prove that W W D(G) is a subspace of W CW (G), the vector space of weight functions w such that G is w-well-covered. We provide a polynomial characterization of W W D(G) for the case that G does not contain cycles of lengths 4, 5, and 6.

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“…The dimension of W CW (G) is denoted by wcdim(G) [3]. More recent results about wcdim can be found in [1] and [2].…”

Section: Andmentioning

confidence: 99%