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5. Linear separation of connected dominating sets in graphs : (extended abstract)Nina Chiarelli, Martin Milanič, 2014, published scientific conference contribution Keywords: povezana dominantna množica, hereditarni grafovski razred, dualno Spernerjev hipergraf, pragovni hipergraf, connected dominating set, hereditary graph class, dually Sperner hypergraph, treshold hypergraph Published in RUP: 15.10.2015; Views: 3406; Downloads: 114 Link to full text |
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7. A note on domination and independence-domination numbers of graphsMartin Milanič, 2013, published scientific conference contribution 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 ▫$\gamma^i < \gamma$▫. Keywords: Vizing's conjecture, domination number, independence-domination number, weakly chordal graph, NP-completeness, hereditary graph class, IDD-perfect graph Published in RUP: 15.10.2013; Views: 3113; Downloads: 128 Full text (300,57 KB) |