On the Eternal Domination Number of Planar Graphs withDiameter 2

An eternal dominating set of a graph is a dominating set $$ D $$ on which mobile guards are initially located (at most one guard is allowed on any vertex). For any infinite sequence of attacks occurring sequentially at vertices, the set $$ D $$ can be modified by moving the guard from an adjacent vertex to the attacked vertex, provided the attacked vertex has no guard on it at the time it is attacked. The configuration of guards after each attack must induce a dominating set. The eternal domination number of a graph is the cardinality of its minimum eternal dominating set. We prove that the eternal domination number of any planar graph of diameter 2 is equal to its clique covering number.