On Domination, 2-Domination, and Italian Domination Numbers

Nader Jafari Rad | M. Hajibaba

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URL :   http://research.shahed.ac.ir/WSR/WebPages/Report/PaperView.aspx?PaperID=116832
Date :  2019/09/13
Publish in :    Utilitas Mathematica

Link :  http://www.combinatorialmath.ca/Utilitas/UM111.html
Keywords :2-dominating set, Italian dominating function

Abstract :

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A set S subseteq V in a graph G=(V,E) is called a textitdominating set if NS = V. The textitdomination number gamma(G) of G is the minimum cardinality of a dominating set in G. A subset Ssubseteq V is a textit2-dominating set if every vertex of V-S has at least two neighbors in S. The minimum cardinality amongst all 2-dominating sets of G is the textit2-domination number, gamma_2(G). An textitItalian dominating function (IDF) on a graph G=(V,E) is a function f :V longrightarrow 0, 1, 2 satisfying the property that for every vertex v in V, with f(v)=0, sum_u in N(v)f(u)geq2. The weight of an IDF f is the value w(f)=f (V) = sum_uin V f(u). The minimum weight of an IDF on a graph G is called the textitItalian domination number of G, denoted by gamma_I(G). In this paper we study graphs with equal domination number and 2-domination number as well as graphs with equal domination number and Italian domination number. In particular, we answer problems posed in Discrete Math. 308 (2008), 2277--2281 and Discrete Appl. Math. 204 (2016), 22--28.