(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let G = (V(G), E(G)) be a simple undirected graph. A function f:V(G)→ {0,1,2} is a super hop Roman dominating function (SHRDF) on G if for every vV(G) with f(v)=0, there exist w, u V(G) with f(w) = 2 and f(u) e 0 such that d_G(v,w)=2, and N_G^2(u){x V(G): f(x) =0}={ v}. The weight of SHRDF f, denoted ω_G^shR(f), is given by ω_G^shR(f)=_y V(G)f(y). The super hop Roman domination number of a graph G, denoted γ_shR(G), is the minimum weight of an SHRDF on G, that is, γ_shR(G)=min{ω_G^shR(f): f is an SHRDF on G }. In this paper, we make an initial investigation of this newly defined variation of hop Roman domination in graphs. Some bounds and exact values of the parameter are obtained and some characterizations on some classes of graphs are given.