Restrained-isolate domination in graphs

A dominating set S ⊆ V (G) of a simple graph G is called a restrained-isolate dominating set of G if the subgraph 〈S〉 induced by S has an isolated vertex and the subgraph 〈V (G)\S〉 induced by V (G)\S has no isolated vertex. A restrained-isolate dominating set of minimum cardinality is called restrained-isolate domination number of G, denoted by γ(r
₀
) (G). In this paper, we characterized the upper and lower bounds of the restrained-isolate domination. We also obtained some characterization of the restrained-isolate domination in some binary operations.