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On The Global Roman Domination Number In Graphs

Hossein Abdollahzadeh Ahangar
Published 2016 · Mathematics

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A Roman dominating function $$ f $$f on a graph $$ G $$G is a global Roman dominating function on $$ G $$G, if $$ f $$f is also a Roman dominating function on $$ \bar{G} $$G¯. The weight of a global Roman dominating function $$ f $$f is the value $$ w(f) = \sum\nolimits_{x \in V(G)} {f(x)} $$w(f)=∑x∈V(G)f(x). The minimum weight of a global Roman dominating function on a graph $$ G $$G is called the global Roman domination number $$ \gamma_{gR} (G) $$γgR(G) of $$ G $$G. In this paper, we present upper bounds for $$ \gamma_{gR} (G) $$γgR(G) in terms of order, diameter, and girth. We give necessary and sufficient conditions for a graph $$ G $$G with property $$ \gamma_{gR} (G) = \gamma_{g} (G) + i $$γgR(G)=γg(G)+i for all $$ i = 0,1, 2,3 $$i=0,1,2,3, where $$ \gamma_{g} (G) $$γg(G) is the global domination number of $$ G $$G. We also describe all connected unicyclic graphs $$ G $$G for which $$ \gamma_{gR} \left( G \right) - \gamma_{R} (G) $$γgRG-γR(G) is maximum.
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