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

Abstract

Let \(G = (V,E)\) be a graph of order \(n\). For \(S \subseteq V(G)\), the set \(N_e(S)\) is defined as the external neighborhood of \(S\) such that all vertices in \(V(G)\backslash S\) have at least one neighbor in \(S\). The differential of \(S\) is defined to be \(\partial(S)=|N_e(S)|-|S|\), and the 2-packing differential of a graph is defined as

$$\partial_{2p}(G) =\max\{\partial(S)\colon S \subseteq V(G) \text{ is a 2-packing}\}.$$

A function \(f\colon V(G) \to \{0,1,2\}\) with the sets \(V_0,V_1,V_2\), where

$$V_i =\{v\in V(G)\colon f(v) = i\},\qquad i \in \{0,1,2\},$$

is a unique response Roman dominating function if \(x \in V_0 \) implies that \(| N( x ) \cap V_2 | = 1\) and \(x \in V_1 \cup V_2 \) implies that \(N( x ) \cap V_2 = \emptyset\). The unique response Roman domination number of \(G\), denoted by \(\mu_R(G)\), is the minimum weight among all unique response Roman dominating functions on \(G\). Let \(\bar{G}\) be the complement of a graph \(G\). The complementary prism \(G\bar {G}\) of \(G\) is the graph formed from the disjoint union of \(G\) and \(\bar {G}\) by adding the edges of a perfect matching between the respective vertices of \(G\) and \(\bar {G}\). The present paper deals with the computation of the 2-packing differential and the unique response Roman domination of the complementary prisms \(G\bar {G}\) 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 \(\partial_{2p} ( G\bar G)\) and \(\mu _R(G\bar G)\) are small are characterized.