m-ETERNAL TOTAL BONDAGE NUMBER IN CIRCULANT GRAPHS

m-ETERNAL TOTAL BONDAGE NUMBER IN CIRCULANT GRAPHS

P. R. L. Pushpam, P. Shanthi

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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).

Keywords

Eternal total domination, total domination, Bondage Number, m- Eternal total Bondage Number