Total Domination Versus Domination in Cubic Graphs
A dominating set in a graph G is a set S of vertices of G such that every vertex not in S has a neighbor in S. Further, if every vertex of G has a neighbor in S, then S is a total dominating set of G. The domination number,γ(G), and total domination number, γ_t(G), are the minimum cardinalities of a dominating set and total dominating set, respectively, in G. The upper domination number, \Gamma(G), and the upper total domination number, \Gamma_t(G), are the maximum cardinalities of a minimal dominating set and total dominating set, respectively, in G. It is known that γ_t(G)/γ (G)≤2 and \Gamma_t(G)/ \Gamma(G)≤2 for all graphs G with no isolated vertex. In this paper we characterize the connected cubic graphs G satisfying γ_t(G)/γ (G)=2, and we characterize the connected cubic graphs G satisfying \Gamma_t(G)/ \Gamma(G)=2.
Joanna Cyman, Magda Dettlaff, Michael A. Henning, Magdalena Lemańska, Joanna Raczek. (2018). Total Domination Versus Domination in Cubic Graphs, 34(1), 261-276. https://doi.org/10.1007/s00373-017-1865-5
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