Well-dominated graphs without cycles of lengths 4 and 5

Vadim E. Levit, David Tankus

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

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 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.

Original languageEnglish
Pages (from-to)1793-1801
Number of pages9
JournalDiscrete Mathematics
Volume340
Issue number8
DOIs
StatePublished - 1 Aug 2017

Keywords

  • Maximal independent set
  • Minimal dominating set
  • Vector space
  • Well-covered graph
  • Well-dominated graph

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