Abstract
The Eternal dominating set of a graph is defined as a set of guards located at vertices, required to protect the vertices of the graph against infinitely long sequences of attacks, such that the configuration of guards induces a dominating set at all times. The eternal m-security number is defined as the minimum number of guards to handle an arbitrary sequence of single attacks using multiple-guard shifts. Kloster-meyer and Mynhardt introduced the concept of an m-eternal total dominating set, which further requires that each vertex with a guard to be adjacent to a vertex with a guard. They defined the m-eternal total domination number of a graph G denoted by γmt∞(G) as the minimum number of guards to handle an arbitrary sequence of single attacks using multiple guard shifts and the configuration of guards always induces a total dominating set. In this paper we examine the effects on γmt∞(G) when G is modified by deleting a vertex which is a non support vertex.
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