BOUNDS ON THE DOUBLE ITALIAN DOMINATION NUMBER OF A GRAPH

Nader Jafari Rad | ازوین

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URL :   http://research.shahed.ac.ir/WSR/WebPages/Report/PaperView.aspx?PaperID=137640
Date :  2021/05/23
Publish in :    Discussiones Mathematicae Graph Theory

Link :  https://www.dmgt.uz.zgora.pl/publish/inpress.php
Keywords :Italian domination, double Italian domination, probabilistic methods.

Abstract :

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For a graph G, a Roman f3g-dominating function is a function f : V 􀀀 f0; 1; 2; 3g having the property that for every vertex u 2 V , if f(u) 2 f0; 1g, then f(Nu)  3. The weight of a Roman f3g-dominating function is the sum w(f) = f(V ) = P v2V f(v), and the minimum weight of a Roman f3g-dominating function is the Roman f3g-domination number, denoted by fR3g(G). In this paper, we present a sharp lower bound for the double Italian domination number of a graph, and improve previous bounds given in D.A. Mojdeh and L. Volkmann, Roman f3g-domination (double Italian domination), Discrete Appl. Math. (2020), in press. We also present a probabilistic upper bound for a generalized version of double Italian domi- nation number of a graph, and show that the given bound is asymptotically best possible.