2-Distance paired-dominating number of graphs

Let (G=(V,E)) be a simple graph without isolated vertices. For a positive integer (k) , a subset (D) of (V(G)) is a (k) -distance paired-dominating set if each vertex in (Vsetminus {D}) is within distance (k) of a vertex in (D) and the subgraph induced by (D) contains a perfect matching. In this paper, we give some upper bounds on the 2-distance paired-dominating number in terms of the minimum and maximum degree, girth, and order.