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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
Citations:
Zbl 1287.05107
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References:
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