Eternal Domination and Clique Covering
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We study the relationship between the eternal domination number of a graph and its clique covering number. Using computational methods, we show that the smallest graph having its eternal domination number less than its clique covering number has 10 vertices. This answers a question of Klostermeyer and Mynhardt [Protecting a graph with mobile guards, Appl. Anal. Discrete Math. 10 (2016), no. 1, 1-29]. We also determine the complete set of 10-vertex and 11-vertex graphs having eternal domination numbers less than their clique covering numbers. In addition, we study the problem on triangle-free graphs, circulant graphs, planar graphs and cubic graphs. Our computations show that all triangle-free graphs and all circulant graphs of order 12 or less have eternal domination numbers equal to their clique covering numbers, and exhibit 13 triangle-free graphs and 2 circulant graphs of order 13 which do not have this property. Using these graphs, we describe a method to generate an infinite family of triangle-free graphs and an infinite family of circulant graphs with eternal domination numbers less than their clique covering numbers. Our computations also show that all planar graphs of order 11 or less, all 3-connected planar graphs of order 13 or less and all cubic graphs of order less than 18 have eternal domination numbers equal to their clique covering numbers. Finally, we show that for any integer k ≥ 2 there exist infinitely many graphs having domination number and eternal domination number equal to k containing dominating sets which are not eternal dominating sets. This answers another question of Klostermeyer and Mynhardt [Eternal and Secure Domination in Graphs, Topics in domination in graphs, Dev. Math. 64 (2020), 445-478, Springer, Cham].
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