On the existence of independent (1,k) -dominating sets for k∈{1,2} in two products of graphs

Abstract

A subset \(J\) of vertices is said to be a \((1,k)\)-dominating set if every vertex \(v\) not belonging to the set \(J\) has a neighbour in \(J\) and there exists also another vertex in \(J\) within the distance at most \(k\) from \(v\). In this paper, we study the problem of the existence of independent \((1,k)\)-dominating sets for \(k\in\{1,2\}\) in the tensor product and in the strong product of two graphs. We give complete characterisations of these graph products, which have independent \((1,1)\)-dominating sets or independent \((1,2)\)-dominating sets, with respect to the properties of their factors.