Graphs with Large Hop Roman Domination Number

Nader Jafari Rad | E.Shabani | A. Poureidi

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URL :   http://research.shahed.ac.ir/WSR/WebPages/Report/PaperView.aspx?PaperID=105909
Date :  2019/01/02
Publish in :    Computer Science Journal of Moldova

Link :  https://web.b.ebscohost.com/abstract?direct=true&profile=ehost&scope=site&authtype=crawler&jrnl=15614042&AN=136553583&h=qZi2fdRGd4jZi9ae9mq0K2bX3C8TznTt5CmG2bMuJ3Xe9GmtE27XTEhIMV82YfE14X7JCTwykumAC1zS0sp8u79FA3d3d&crl=c&resultNs=AdminWebAuth&resultLocal=ErrCrlNotAuth&crlhashurl=login.aspx3fdirect3dtrue26profile3dehost26scope3dsite26authtype3dcrawler26jrnl3d1561404226AN3d136553583
Keywords :Graphs, Domination, Number,Function

Abstract :

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A subset S of vertices of a graph G is a hop dominating set if every vertex outside S is at distance two from a vertex of S. A Roman dominating function on a graph G = (V,E) is a function f : V (G) −→ 0, 1, 2 satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. A hop Roman dominating function (HRDF) of G is a function f : V (G) −→ 0, 1, 2 having the property that for every vertex v ∈ V with f(v) = 0 there is a vertex u with f(u) = 2 and d(u, v) = 2. The weight of a HRDF f is the sum f(V ) = Pv2V f(v). The minimum weight of a HRDF on G is called the hop Roman domination number of G and is denoted by hR(G). In this paper we characterize all graphs G of order n with hR(G) = n or hR(G) = n − 1.