Note on the Eternal Domination Number of Planar Graphs and Vertex-Critical Graphs

An eternal dominating set S of G is a dominating set with the following property: for any infinite sequence of selecting vertices, called attacks, moving one vertex in S along an edge from its current position to the attacked vertex results in another dominating set of G.The size of a minimum eternal dominating set of G is called the eternal domination number of G, denoted by (G).Klostermeyer and Mynhardt [Appl.Anal.Discrete Math. 10 (2016) 1-29] asked whether every planar graph G satisfies (G) = (G), where (G) is the clique covering number of G. Further, MacGillivray, Mynhardt and Virgile [Electron.J. Graph Theory Appl. 10 (2022) 603-624] showed that a smallest planar graph G with (G) < (G) is 2-connected and vertex-critical with respect to (G) if such a graph exists.In this paper, we give some results for the eternal domination number of planar graphs and vertex-critical graphs.Moreover, we verify the existence of graphs G with (G) < (G) for every non-spherical surface.