(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(u)=3. In addition, if every subgraph induced by the set {vV(G):f(v)>0} is an isolated free, then we say that f is a total modern Roman dominating function of G. The minimum weight ω_G^tmR(f)=_vV(G)f(v) of a total modern Roman dominating function f of G is called the total modern Roman domination number γ_tmR(G) of G. In this paper, we initiate the study of total modern Roman domination. We characterize graphs with smaller total modern Roman domination number and obtain the γ_tmR(G) of some special graphs. Moreover, we investigate and characterize the total modern Roman domination of the join and corona of graphs.