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Domination versus independent domination in cubic graphs. (English) Zbl 1277.05129
Summary: A set \(S\) of vertices in a graph \(G\) is a dominating set if every vertex not in \(S\) is adjacent to a vertex in \(S\). If, in addition, \(S\) is an independent set, then \(S\) is an independent dominating set. The domination number \(\gamma (G)\) of \(G\) is the minimum cardinality of a dominating set in \(G\), while the independent domination number \(i(G)\) of \(G\) is the minimum cardinality of an independent dominating set in \(G\). In this paper we show that if \(G\neq K(3,3)\) is a connected cubic graph, then \(i(G)/\gamma (G)\leq 4/3\). This answers a question posed by W. Goddard et al. in [Ann. Comb. 16, No. 4, 719–732 (2012; Zbl 1256.05169)] where the bound of \(3/2\) is proven. In addition we characterize the graphs achieving this ratio of \(4/3\).

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
Zbl 1256.05169
Full Text: DOI
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