Triple connected eternal domination in graphs

Abstract

The concept of Triple connected domination number was introduced by G. Mahadevan et. al., in [10]. The concept of eternal domination in graphs was introduced by W. Goddard., et. al., in [3]. The dominating set S0(⊆ V (G)) of the graph G is said to be an eternal dominating set, if for any sequence v1, v2, v3, . . . vk of vertices, there exists a sequence of vertices u1, u2, u3, . . . uk with ui ∈ Si−1 and ui equal to or adjacent to vi, such that each set Si = Si−1−{ui}∪{vi} is dominating set in G. The minimum cardinality taken over the eternal dominating sets in G is called the eternal domination number of G and it is denoted by γ∞(G). In this paper we introduce another new concept Triple connected eternal domination in graph. The eternal dominating set S0(⊆ V (G)) of the graph G is said to be a triple connected eternal dominating set, if each dominating set Si is triple connected. The minimum cardinality taken over the triple connected eternal dominating sets in G is called the triple connected eternal domination number of G and it is denoted by γtc,∞(G). We investigate this number for some standard graphs and obtain many results with other graph theoretical parameters.