TRIPLE CONNECTED ETERNAL DOMINATION IN GRAPHS

TRIPLE CONNECTED ETERNAL DOMINATION IN GRAPHS

G. Mahadevan, T. Ponnuchamy, S. Avadayappan

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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 cardinal- ity 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.

Keywords

Triple connected domination number, Eternal domination in graphs, Triple connected eternal domination number of graphs.