TY - JOUR

T1 - Well-dominated graphs without cycles of lengths 4 and 5

AU - Levit, Vadim E.

AU - Tankus, David

N1 - Publisher Copyright:
© 2017 Elsevier B.V.

PY - 2017/8/1

Y1 - 2017/8/1

N2 - 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 minimum 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-well-dominated is a vector space, and denote that vector space by WWD(G). We show that WWD(G) is a subspace of WCW(G), the vector space of weight functions w such that G is w-well-covered. We provide a polynomial characterization of WWD(G) for the case that G does not contain cycles of lengths 4, 5, and 6.

AB - 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 minimum 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-well-dominated is a vector space, and denote that vector space by WWD(G). We show that WWD(G) is a subspace of WCW(G), the vector space of weight functions w such that G is w-well-covered. We provide a polynomial characterization of WWD(G) for the case that G does not contain cycles of lengths 4, 5, and 6.

KW - Maximal independent set

KW - Minimal dominating set

KW - Vector space

KW - Well-covered graph

KW - Well-dominated graph

UR - http://www.scopus.com/inward/record.url?scp=85018508026&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2017.02.021

DO - 10.1016/j.disc.2017.02.021

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AN - SCOPUS:85018508026

SN - 0012-365X

VL - 340

SP - 1793

EP - 1801

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 8

ER -