On the Global Double Roman Domination of Graphs

Abstract

We establish some inequalities between the global double Roman domination number $$\gamma _\mathrm{gdR}(G)$$ and double Roman domination number $$\gamma _\mathrm{dR}(G)$$ of graphs G. We also completely characterize the trees T with $$\gamma _\mathrm{gdR}(T)=\gamma _\mathrm{dR}(T)+k$$ for $$k\in \{0,1,2,3\}$$, which partially answer an open problem posed by Shao et al. (J Discrete Math Sci Cryptogr 22:31–44, 2019).