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On the ratio of the domination number and the independent domination number in graphs. (English) Zbl 1300.05219
Summary: We let $$\gamma(G)$$ and $$i(G)$$ denote the domination number and the independent domination number of $$G$$, respectively. Recently, N. J. Rad and L. Volkmann [ibid. 161, No. 18, 3087–3089 (2013; Zbl 1287.05107)] conjectured that $$i(G) / \gamma(G) \leq \Delta(G) / 2$$ for every graph $$G$$, where $$\Delta(G)$$ is the maximum degree of $$G$$. In this note, we construct counterexamples of the conjecture for $$\Delta(G) \geq 6$$ and give a sharp upper bound of the ratio $$i(G) / \gamma(G)$$ by using the maximum degree of $$G$$.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C07 Vertex degrees 05C35 Extremal problems in graph theory
Zbl 1287.05107
Full Text:
##### References:
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