2-HOP ROMAN DOMINATION OF SOME PRODUCT GRAPHS

Abstract

In this paper, we  introduce the study of 2-hop Roman dominating functions of some product graphs. A 2-hop Roman dominating (2HRD) function on a graph \(G=(V,E)\)is a function \(f:V(G)\rightarrow\{0,1,2\}\) such that for every vertex \(v\) with \(f(v)=0\), there are two vertices \(x,y\) with \(f(x)=f(y)=2\) and \(d(v,x)=d(v,y)=2\). The weight of 2HRD function is the sum of its function values over all the vertices. The 2-hop Roman domination number of \(G\) denoted by \(\gamma_{2hR}(G)\) is the minimum weight of a 2HRD function in \(G\). We present the upper bond of 2-hop Roman domination number of Cartesian product graph. Also, we determine the  2-hop Roman domination number of tensor product graphs.