(untitled) (European Journal of Pure and Applied Mathematics, 2023)
Given a simple graph G=(V(G),E(G)), a function f:V(G)→ {0,1,2} is a hop Italian dominating function if for every vertex v with f(v)=0 there exists a vertex u with f(u)=2 for which u and v are of distance 2 from each other or there exist two vertices w and z for which f(w)=1=f(z) and each of w and z is of distance 2 from v. The minimum weight _vV(G)f(v) of a hop Italian dominating function is the hop Italian domination number of G, and is denoted by γ_hI(G). In this paper, we initiate the study of the hop Italian domination. In particular, we establish some properties of the the hop Italian dominating function and explore the relationships of the hop Italian domination number with the hop Roman domination number Rad2,Natarajan and with the 2-hop domination number Canoy. We study the concept under some binary graph operations. We establish tight bounds and determine exact values for their respective hop Italian domination numbers.