Relatively prime restrained detour domination number of a graph

Abstract

Consider a connected graph  G=(V, E) with a minimum of two vertices. A subset ⊆ is termed a relatively prime restrained detour dominating set of G if it fulfills two conditions: firstly, S must be a relatively prime detour dominating set, and secondly, the induced subgraph should not contain any isolated vertices. The relatively prime restrained detour domination number, denoted as  γ_rprdn(G), is defined as the minimum cardinality of such a set that satisfies these conditions. Precise values for certain standard graphs, limits and some interesting results are established.