From the Strong Differential to Italian Domination in Graphs

Abstract Given a graph G and subset of vertices $$D\subseteq V(G)$$ D ⊆ V ( G ) , the external neighbourhood D is defined as $$N_e(D)=\{u\in V(G){\setminus } D:\, N(u)\cap D\ne \varnothing \}$$ N e = { u ∈ \ : ∩ ≠ ∅ } where N ( u ) denotes open . Now, given vertex $$v\in D$$ v private v with respect to be $$\mathrm{epn}(v,D)=\{u\in D: \, D=\{v\}\}.$$ epn , . The strong differential set $$\partial _s(D)=|N_e(D)|-|D_w|,$$ ∂ s | - w $$D_w=\{v\in \mathrm{epn}(v,D)\ne In this paper, we focus on study graph, which $$\begin{aligned} \partial _s(G)=\max \{\partial _s(D):\, D\subseteq V(G)\}. \end{aligned}$$ max Among other results, obtain general bounds _s(G)$$ prove Gallai-type theorem, states that _s(G)+\gamma _{_I}(G)=\mathrm{n}(G)$$ + γ I n $$\gamma _{_I}G)$$ Italian domination number Therefore, can see theory in graphs new approach domination. One advantages it allows us without use functions. As expect, derive results graph.