On the $2$-packing differential of a graph

TL;DR

This paper introduces the concept of the 2-packing differential in graphs, explores its relationship with various graph parameters, and establishes a Gallai-type theorem linking it to the unique response Roman domination number, which is shown to be NP-hard to compute.

Contribution

It defines the 2-packing differential, connects it with multiple graph parameters, and proves a new Gallai-type theorem relating it to the unique response Roman domination number.

Findings

The 2-packing differential is closely related to several graph parameters.

A Gallai-type theorem linking the 2-packing differential and the unique response Roman domination number is established.

Computing the unique response Roman domination number is NP-hard.

Abstract

Let $G$ be a graph of order $n(G)$ and vertex set $V(G)$. Given a set $S\subseteq V(G)$, 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 $\partial_{2p}(G)=\max\{\partial(S): S\subseteq V(G) \text{ is a }2\text{-packing}\}.$ 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 $\partial_{2p}(G)+\mu_{_R}(G)=n(G)$, 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.