On proper (1,2)-dominating sets in graphs

In 2008, Hedetniemi et al. introduced the concept of (1, k)-domination and obtained some interesting results for (1, 2)-domination. Obviously every (1, 1)-dominating set of a graph (known as 2- dominating set) is (1, 2)-dominating; to distinguish these concepts, we define a proper (1, 2) -dominating set of a graph as follows: a subset D is a proper (1, 2)-dominating set of a graph if D is (1, 2)-dominating and it is not a (1, 1) -dominating set of a graph. In particular, we introduce proper (1, 2)-domination parameters and study their relations with (1, 2) -domination parameters and the classical domination numbers.