m-Eternal total bondage number in circulant graphs

Abstract

An 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. Klostermeyer and Mynhardt defined the m-eternal total domination number of a graph G denoted by γ∞mt(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. We define the m-Eternal Total bondage number of a graph G denoted by bmt(G) as the minimum cardinality of set of edges E′⊆ E(G) for which γ∞mt(G − E′) > γ∞mt(G) and G − E′ does not contain isolated vertices. In this paper we find the exact values of bmt(G) for Circulant graphs Cn(1, 2) and Cn(1, 3).