From the strong differential to Italian domination in graphs

TL;DR

This paper introduces the concept of strong differential in graphs, establishes bounds, and proves a Gallai-type theorem linking it to the Italian domination number, offering a new approach to studying Italian domination without functions.

Contribution

It defines the strong differential of a graph, proves a Gallai-type theorem relating it to Italian domination, and provides new bounds and results for Italian domination.

Findings

Established bounds on the strong differential of graphs.

Proved a Gallai-type theorem connecting strong differential and Italian domination.

Derived new results on the Italian domination number.

Abstract

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.