(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let G=(V(G), E(G)) be any connected graph. A function f:V(G) → {0,1,2,3} is a modern Roman dominating function of G if for each vV(G) with f(v)=0, there exist u,w N_G (v) such that f(u)=2 and f(w)=3; and for each vV(G) with f(v)=1, there exists u N_G (v) such that f(u)=2 or f(w)=3. The weight of a modern Roman dominating function f of G is the sum ω_G^mR(f)=_vV(G)f(v) and its minimum weight is called the modern Roman domination number γ_mR(G) of G. In this paper, we characterize graphs with smaller modern Roman domination number and obtain the γ_mR(G) of some special graphs. Moreover, we investigate and characterize the modern Roman domination of the join and corona of graphs.