2-Distance paired-dominating number of graphs

Abstract

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 $V\setminus {D}$ 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.