Restrained dr-Power Dominating Sets in Graphs

Consider a nontrivial connected graph G. In this context, a set R that is not empty and a subset of V (G) is referred to as a restrained dr-power dominating set of G. This means that the induced subgraph of the complement of R in G does not contain any isolated vertex and qualifies as a dr-power dominating set of G. To determine the restrained dr-power domination number of G, denoted as y*rpw (G), we look at the minimum cardinality of a restrained dr-power dominating set. This study presents significant insights into the restrained dr-power dominating set of a graph G. It provides concrete realizations and exact values for the restrained dr-power domination number within specific graph classes, such as path and cycle graphs, as well as in the context of join and corona operations. Additionally, characterizations of the restrained dr-power dominating set in the join and corona of graphs are demonstrated.