Restrained Roman domination in complementary prisms from restrained differential prospect

For a graph [Formula: see text], a restrained Roman dominating function [Formula: see text] has the property that every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text] and at least one vertex [Formula: see text] for which [Formula: see text]. The weight of a restrained Roman dominating function is the sum [Formula: see text]. The minimum weight of a restrained Roman dominating function is called the restrained Roman domination number and is denoted by [Formula: see text]. Let the external neighborhood set [Formula: see text] of a set [Formula: see text] be the set of vertices in [Formula: see text] that have a neighbor in the vertex set [Formula: see text]. The restrained external neighborhood of [Formula: see text] is defined as [Formula: see text]. The restrained differential of [Formula: see text] is defined as [Formula: see text] and the restrained differential of a graph is defined as [Formula: see text]}. 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 [Formula: see text] of [Formula: see text] arises from the disjoint union of [Formula: see text] and its complement [Formula: see text] by adding the edges of a perfect matching between the corresponding vertices of [Formula: see text] and [Formula: see text]. This paper is devoted to the computation of restrained differential and restrained Roman domination of complementary prisms, and results are obtained for complementary prims of specified family of graphs.