Secure Triple Connected Domination Number of a Graph (Journal of Informatics and Mathematical Sciences, 2017)
Secure domination is a well-studied concept [3, 4, 5]. In this domination, a vertex outside has the chance of coming inside the dominating set by replacing an element of the set without affecting domination. This idea is combined with the concept of triple connected domination, by considering a path between any three vertices of a graph [10, 11, 12], to introduce a new parameter called secure triple connected domination. A secure dominating set \(S\) of \(V\) of a nontrivial graph \(G\) is said to be secure triple connected dominating set, if the induced sub graph \(\langle S\rangle\) is triple connected. Among all the secure triple connected dominating sets of the graph \(G\), a set having the minimum cardinality is called the secure triple connected domination number denoted by \(γ_{stc}\) of \(G\). We have determined the exact values of secure triple connected domination number for some standard graphs and obtained bounds for this new parameter. NORDHAUS-GADDUM type results and the relationship of this parameter with other graph theoretical parameters are also discussed.
Keywords: Domination number; Secure domination number; Triple connected domination number; Secure connected domination number; Secure triple connected dominating set; Secure triple connected domination number