Eternally Secure Sets, Independence Sets and Cliques
William F. Klostermeyer, Gary MacGillivray, S. Arumugam · AKCE International Journal of Graphs and Combinatorics · 2005
Goddard, Hedetniemi, and Hedetniemi conjectured that if the independence number of a graph is equal to the eternal security number, then the independence number is equal to the chromatic number of the complement of the graphs (a.k.a, the clique covering number of the graph) [JCMCC, vol. 52, pp. 160-180]. We prove the conjecture is true when the independence number is two and provide counterexamples when the independence number is greater than two.