## TRIPLE CONNECTED ETERNAL DOMINATION IN GRAPHS

## TRIPLE CONNECTED ETERNAL DOMINATION IN GRAPHS

*G. Mahadevan, T. Ponnuchamy, S. Avadayappan*

[**PDF**]

## 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.