Abstract
An Italian dominating function (IDF) on a graph $G=(V,E)$ is a function $f:V\rightarrow \{0,1,2\}$ satisfying the condition that for every vertex $v\in V$ with $f(v)=0$ , either $v$ is adjacent to a vertex assigned 2 under $f$ , or $v$ is adjacent to at least two vertices assigned 1 under $f$ . The weight of an IDF $f$ is the value $\sum _{v\in V}f(v)$ . The Italian domination number of a graph $G$ is the minimum weight of an IDF on $G$ . The Italian reinforcement number of a graph is the minimum number of edges that have to be added to the graph in order to decrease the Italian domination number. In this paper, we initiate the study of Italian reinforcement number and we present some sharp upper bounds for this parameter. In particular, we determine the exact Italian reinforcement numbers of some classes of graphs.
Highlights
Let G be a simple graph with vertex set V (G) and edge set E(G)
If the graph G is clear from the context, we will write N (v), N [v], N (S), d(v), and pn(v, S) rather than NG(v), NG[v], NG(S), dG(v), (G) and pnG(v, S), respectively
An Italian dominating function (IDF) on a graph G is defined as a function f : V (G) → {0, 1, 2} satisfying the condition that for every vertex v ∈ V (G) with f (v) = 0, f (N (v)) ≥ 2, that is, either there is a vertex u ∈ N (v) with f (u) = 2, or at least two vertices x, y ∈ N (v) with f (x) = f (y) = 1
Summary
Let G be a simple graph with vertex set V (G) and edge set E(G). The open neighborhood of a vertex v in G is the set NG(v) = {u ∈ V (G) : uv ∈ E(G)} and its closed neighborhood is the set NG[v] = NG(v) ∪ {v}. Since the domination number of every graph G is at least 1, by convention Kok and Mynhardt defined r(G) = 0 if γ (G) = 1. An Italian dominating function (IDF) on a graph G is defined as a function f : V (G) → {0, 1, 2} satisfying the condition that for every vertex v ∈ V (G) with f (v) = 0, f (N (v)) ≥ 2, that is, either there is a vertex u ∈ N (v) with f (u) = 2, or at least two vertices x, y ∈ N (v) with f (x) = f (y) = 1. The Italian domination number of a graph G, denoted by γI (G), is the minimum weight of an IDF on G. The Italian reinforcement number of a graph G, denoted by rI (G), is the minimum size of an IR-set of G. We derive some sharp upper bounds on the Italian reinforcement number and we determine exact values of Italian reinforcement number of some classes of graphs
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