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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 \(G\).

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: DOI
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