Independent Perfect Secure Domination in Certain Cartesian Product Graphs - Thomas, Merlin; Cynthia, V. Jude Annie (Communications in Mathematics and Applications, 2025)
In a graph \(G=(V(G),E(G))\), a set \(S\subseteq V(G)\) is a dominating set of \(G\) when every vertex in \(V(G)\setminus S\) is adjacent to at least one vertex in \(S\). A dominating set \(S\) is called an independent perfect secure dominating set of \(G\) if \(S\) is an independent set and if for each vertex \(w\in V(G)\setminus S\), there exists a unique vertex \(v\in S\) such that \(v\) is adjacent to \(w\) and \((S\setminus\{v\})\cup\{w\}\) is a dominating set of \(G\). The minimum cardinality of an independent perfect secure dominating set of \(G\) is called the independent perfect secure domination number of \(G\). In this paper, we determine the exact value of the independent perfect secure domination number of \(G\square H\), where \(G\) is a complete graph and \(H\) belongs to specific graph classes such as complete graphs, complete bipartite graphs, paths, and cycles. Upper bounds for the independent perfect secure domination number of certain other Cartesian product graphs are also dealt with in this paper.