# On The Global Roman Domination Number In Graphs

Published 2016 · Mathematics

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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