Let G be a graph of order \({\text {n}}(G)\) 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 \(\partial _{2p}(G)+\mu _{_R}(G)={\text {n}}(G)\), 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.

令G为阶\({\text {n}}(G)\)和顶点集V ( G ) 的图。给定一个集合\(S\subseteq V(G)\)，我们将S的外部邻域定义为\(V(G){\setminus } S\)中所有顶点的集合\(N_e(S) \)在S中至少有一个邻居。S的微分定义为\(\partial (S)=|N_e(S)|-|S|\)。在本文中，我们介绍了对图的 2-packing 微分的研究，我们将其定义为\(\partial _{2p}(G)=\max \{\partial (S):\, S\subseteq V (G) \text { 是一个 }2\text {-packing}\}.\)我们表明，2-packing 微分与几个图参数密切相关，包括包装数、独立支配数、总支配数、完美微分和唯一响应罗马支配数。特别是，我们表明 2-packing 微分理论是一个合适的框架，可以在不使用函数的情况下研究图的唯一响应罗马支配数。在其他结果中，我们得到了一个 Gallai 型定理，它指出\(\partial _{2p}(G)+\mu _{_R}(G)={\text {n}}(G)\)，其中\（\亩_ {_ R}（G）\）表示的独特的反应罗马控制数ģ。作为研究的结果，我们在\(\mu _{_R}(G)\)，我们表明找到这个参数的问题是 NP-hard。此外，还讨论了词典产品图的特殊情况。