Research Information System
Discrete Mathematics, Algorithms and Applications, 2025 (ESCI, Scopus)
For a graph G = (V,E), a restrained Roman dominating function f : V →{0, 1, 2} has the property that every vertex v with f(v) = 0 is adjacent to at least one vertex u for which f(u) = 2 and at least one vertex w for which f(w) = 0. The weight of a restrained Roman dominating function is the sum f(V ) =∑v∈Vf(v). The minimum weight of a restrained Roman dominating function is called the restrained Roman domination number and is denoted by γrR(G). Let the external neighborhood set Se of a set S ⊆ V (G) be the set of vertices in V ∖S that have a neighbor in the vertex set S. The restrained external neighborhood of S is defined as Sr = {v ∈ Se : N(v) ∩ Se≠∅}. The restrained differential of S is defined as ∂r(S) = |Sr|−|S| and the restrained differential of a graph is defined as ∂r(G) =max{∂r(S) : S ⊆ V}. The theory of restrained differential is perfectly integrated into the theory of restrained Roman domination for development by the use of a Gallai-type theorem proven recently. The complementary prism GḠ of G arises from the disjoint union of G and its complement Ḡ by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. This paper is devoted to the computation of restrained differential and restrained Roman domination of complementary prisms, and results are obtained for complementary prisms of specified family of graphs.