On the 2-Packing Differential of a Graph

Identifier:  imarina:9227041

Handle: https://hdl.handle.net/20.500.11797/imarina9227041

Authors:  Martínez, AC; Puertas, ML; Rodríguez-Velázquez, JA

Abstract:
Let G be a graph of order n(G) and vertex set V(G). Given a set S subset of V (G), we define the external neighbourhood of S as the set Ne(S) of all vertices in V (G)\S having at least one neighbour in S. The differential of S is defined to be partial derivative(S) = vertical bar N-e(S)vertical bar - vertical bar S vertical bar. In this paper, we introduce the study of the 2-packing differential of a graph, which we define as partial derivative(2p)(G) = max{partial derivative(S) : S subset of V(G) is a 2-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 derivative(2p)(G)+ mu(R)(G) = 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.