Abstract
An Italian dominating function of G is a function f : V ( G ) → { 0 , 1 , 2 } , for every vertex v such that f ( v ) = 0 , it holds that ∑ u ∈ N ( v ) f ( u ) ≥ 2 . The Italian domination number γ I ( G ) is the minimum weight of an Italian dominating function on G. In this paper, we determine the exact values of the Italian domination numbers of P ( n , 3 ) .
Highlights
In a graph G with vertex set V ( G ) and edge set E( G ), for any v ∈ V, the open neighborhood of a vertex v ∈ V is a set {u|(u, v) ∈ E}, denoted by N (v) and the closed neighborhood of a vertex v ∈ V is N [v] = N (v) ∪ {v}
A set D ⊆ V, if for every vertex v ∈ V \ D, v is adjacent to a vertex in
The Roman domination number is the minimum weight of an Roman dominating function (RDF), denoted by γR ( G )
Summary
The minimum cardinality of a domination set of G is called the domination number of G, denoted by γ( G ). The Roman domination number is the minimum weight of an RDF, denoted by γR ( G ). The Italian domination number is the minimum weight of an IDF on G, denoted as γ I (G). Mathematics 2019, 7, 714 demonstrate that the Italian domination number is equal to the 2-rainbow domination number for trees and cactus graphs with no even cycles. Henning et al [10] characterize the tree graphs T. obtain the Italian domination numbers of prove γ II(Cnn2C55) = 2n. It is worthwhile to study the Italian domination numbers for some special classes of graphs.
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