Geodesic
Domination, Eternal Domination, and Clique Covering
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 283-300

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Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique covering numbers of graph are explored in this paper. Among other results, we characterize bipartite and triangle-free graphs with domination and eternal domination numbers equal to two, trees with equal m-eternal domination and clique covering numbers, and two classes of graphs with equal domination, eternal domination and clique covering numbers.
Keywords: dominating set, eternal dominating set, independent set, clique cover 
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Klostermeyer, William F.; Mynhardt, C.M. Domination, Eternal Domination, and Clique Covering. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 283-300. http://geodesic.mathdoc.fr/item/DMGT_2015_35_2_a7/