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Abstract: Vizing's conjecture is true for graphs ▫$G$▫ satisfying ▫$\gamma^i(G) = \gamma(G)$▫, where ▫$\gamma(G)$▫ is the domination number of a graph ▫$G$▫ and ▫$\gamma^i(G)$▫ is the independence-domination number of ▫$G$▫, that is, the maximum, over all independent sets ▫$I$▫ in ▫$G$▫, of the minimum number of vertices needed to dominate ▫$I$▫. The equality ▫$\gamma^i(G) = \gamma(G)$▫ is known to hold for all chordal graphs and for chordless cycles of length ▫$0 \pmod{3}$▫. We prove some results related to graphs for which the above equality holds. More specifically, we show that the problems of determining whether ▫$\gamma^i(G) = \gamma(G) = 2$▫ and of verifying whether ▫$\gamma^i(G) \ge 2$▫ are NP-complete, even if ▫$G$▫ is weakly chordal. We also initiate the study of the equality ▫$\gamma^i = \gamma$▫ in the context of hereditary graph classes and exhibit two infinite families of graphs for which ▫$\gamma^i < \gamma$▫.
Found in: ključnih besedah
Summary of found: ...\gamma(G)$▫ is known to hold for all chordal graphs and for chordless cycles of length...
Keywords: Vizing's conjecture, domination number, independence-domination number, weakly chordal graph, NP-completeness, hereditary graph class, IDD-perfect graph
Published: 15.10.2013; Views: 2400; Downloads: 122
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Found in: ključnih besedah
Summary of found: ...graph, forbidden induced subgraph characterization, split graph, chordal graph, minimal cutset, minimal separator, 1-Sperner hypergraph,...
Keywords: connected dominating set, connected domination, connected-domishold graph, forbidden induced subgraph characterization, split graph, chordal graph, minimal cutset, minimal separator, 1-Sperner hypergraph, threshold hypergraph, threshold Boolean function, polynomial-time algorithm
Published: 04.04.2019; Views: 1287; Downloads: 156
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