Abstract
A dominating set in a graph \(G\) is a set \(S\) of vertices such that every vertex outside \(S\) has a neighbor in \(S\); the domination number \(\gamma (G)\) is the minimum size of such a set. The independent domination number, written \(i(G)\), is the minimum size of a dominating set that also induces no edges. Henning and Southey conjectured \(i(G)/\gamma (G) \le 6/5\) for every cubic (3-regular) graph \(G\) with sufficiently many vertices. We provide an infinite family of counterexamples, giving for each positive integer \(k\) a 2-connected cubic graph \(H_k\) with \(14k\) vertices such that \(i(H_k)=5k\) and \(\gamma (H_k)=4k\).
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Research supported by Recruitment Program of Foreign Experts, 1000 Talent Plan, State Administration of Foreign Experts Affairs, China.
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O, S., West, D.B. Cubic Graphs with Large Ratio of Independent Domination Number to Domination Number. Graphs and Combinatorics 32, 773–776 (2016). https://doi.org/10.1007/s00373-015-1580-z
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DOI: https://doi.org/10.1007/s00373-015-1580-z