Given a graph G and a subset of vertices \(D\subseteq V(G)\), the external neighbourhood of D is defined as \(N_e(D)=\{u\in V(G){\setminus } D:\, N(u)\cap D\ne \varnothing \}\), where N(u) denotes the open neighbourhood of u. Now, given a subset \(D\subseteq V(G)\) and a vertex \(v\in D\), the external private neighbourhood of v with respect to D is defined to be \(\mathrm{epn}(v,D)=\{u\in V(G){\setminus } D: \, N(u)\cap D=\{v\}\}.\) The strong differential of a set \(D\subseteq V(G)\) is defined as \(\partial _s(D)=|N_e(D)|-|D_w|,\) where \(D_w=\{v\in D:\, \mathrm{epn}(v,D)\ne \varnothing \}\). In this paper, we focus on the study of the strong differential of a graph, which is defined as

Among other results, we obtain general bounds on \(\partial _s(G)\) and we prove a Gallai-type theorem, which states that \(\partial _s(G)+\gamma _{_I}(G)=\mathrm{n}(G)\), where \(\gamma _{_I}G)\) denotes the Italian domination number of G. Therefore, we can see the theory of strong differential in graphs as a new approach to the theory of Italian domination. One of the advantages of this approach is that it allows us to study the Italian domination number without the use of functions. As we can expect, we derive new results on the Italian domination number of a graph.

给定图G和顶点子集\(D\subseteq V(G)\)，D的外部邻域定义为\(N_e(D)=\{u\in V(G){\setminus } D ：\，N（U）\帽d \ NE \ varnothing \} \） ，其中ñ（Û）表示的开口附近ü。现在，给定子集\(D\subseteq V(G)\)和顶点\(v\in D\)，v相对于D的外部私有邻域定义为\(\mathrm{epn}( v,D)=\{u\in V(G){\setminus } D: \, N(u)\cap D=\{v\}\}.\)集合的强微分\(D\ subseteq V(G)\)定义为\(\partial _s(D)=|N_e(D)|-|D_w|,\)其中\(D_w=\{v\in D:\, \mathrm{epn}(v,D)\ne \varnothing \}\)。在本文中，我们重点研究图的强微分，其定义为

在其他结果中，我们获得了\(\partial _s(G)\) 的一般边界，并证明了一个 Gallai 型定理，该定理指出\(\partial _s(G)+\gamma _{_I}(G)= \ mathrm {N}（G）\） ，其中\（\伽马_ {_ I} G）\）表示的意大利控制数ģ。因此，我们可以将图强微分理论视为意大利统治理论的一种新方法。这种方法的优点之一是它允许我们在不使用函数的情况下研究意大利统治数。正如我们所料，我们在图的意大利统治数上得出了新的结果。