Unique Response Roman Domination Versus 2-Packing Differential in Complementary Prisms

Let G = (V, E) be a graph of order n. For S subset of V (G), the set N-e(S) is defined as the external neighborhood of S such that all vertices in V (G)\S have at least one neighbor in S. The differential of S is defined to be partial derivative(S) = |N-e(S)| - |S|, and the 2-packing differential of a graph is defined as partial derivative(2p)(G) = max{partial derivative( S): S subset of V (G) is a 2-packing}. A function f : V (G) -> {0, 1, 2} with the sets V-0, V-1, V-2, where V-i = {v is an element of V (G): f(v) = i}, i is an element of {0, 1, 2}, is a unique response Roman dominating function if x is an element of V-0 implies that | N(x) boolean AND V2 | = 1 and x is an element of V-1 boolean OR V-2 implies that N(x) boolean AND V-2 = (sic). The unique response Roman domination number of G, denoted by mu(R)(G), is the minimum weight among all unique response Roman dominating functions onG. Let (G) over bar be the complement of a graphG. The complementary prism G (G) over bar of Gis the graph formed from the disjoint union of G and (G) over bar by adding the edges of a perfect matching between the respective vertices of G and (G) over bar. The present paper deals with the computation of the 2-packing differential and the unique response Roman domination of the complementary prisms G (G) over bar by the use of a proven Gallai-type theorem. Particular attention is given to the complementary prims of special types of graphs. Furthermore, the graphs G such that partial derivative(2p)(G (G) over bar) and mu(R)(G (G) over bar) are small are characterized.