  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 $epn(v,D)=\{u\in
V(G)\setminus D: N(u)\cap D=\{v\}\}.$ The strong differential of a set
$D\subseteq V(G)$ is defined as $\partial_s(D)=|N_e(D)|-|D_w|,$ where
$D_w=\{v\in D: epn(v,D)\neq \varnothing\}$. In this paper we focus on the study
of the strong differential of a graph, which is defined as $$\partial_s(G)=\max
\{\partial_s(D): D\subseteq V(G)\}.$$ Among other results, we obtain general
bounds on $\partial_s(G)$ and we prove a Gallai-type theorem, which states that
$\partial_s(G)+\gamma_{I}(G)=n(G)$, 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.
