On the 2-Packing Differential of a Graph

Abstract

Let G be a graph of order {text {n}}(G) and vertex set V(G). Given a set Ssubseteq V(G), we define the external neighbourhood of S as the set N_e(S) of all vertices in V(G){setminus } S having at least one neighbour in S. The differential of S is defined to be partial (S)=|N_e(S)|-|S|. In this paper, we introduce the study of the 2-packing differential of a graph, which we define as partial _{2p}(G)=max {partial (S):, Ssubseteq V(G) text { is a }2text {-packing}}. We show that the 2-packing differential is closely related to several graph parameters, including the packing number, the independent domination number, the total domination number, the perfect differential, and the unique response Roman domination number. In particular, we show that the theory of 2-packing differentials is an appropriate framework to investigate the unique response Roman domination number of a graph without the use of functions. Among other results, we obtain a Gallai-type theorem, which states that partial _{2p}(G)+mu _{_R}(G)={text {n}}(G), where mu _{_R}(G) denotes the unique response Roman domination number of G. As a consequence of the study, we derive several combinatorial results on mu _{_R}(G), and we show that the problem of finding this parameter is NP-hard. In addition, the particular case of lexicographic product graphs is discussed.