(untitled) (European Journal of Pure and Applied Mathematics, 2023)
Let G be a connected graph. A function f: V(G)→ {0,1,2} is a geodetic Roman dominating function (or GRDF) if every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2 and V_1 V_2 is a geodetic set in G. The weight of a geodetic Roman dominating function f, denoted by ω_G^gR(f), is given by ω_G^gR(f)=_v V(G)f(v). The minimum weight of a GRDF on G, denoted by γ_gR(G), is called the geodetic Roman domination number of G. In this paper, we give some properties of geodetic Roman domination and determine the geodetic Roman domination number of some graphs.