Upper bounds on the paired-domination number

A set SS of vertices in a graph GG is a paired-dominating set   of GG if every vertex of GG is adjacent to some vertex in SS and the subgraph induced by SS contains a perfect matching. The minimum cardinality of a paired-dominating set of GG is the paired-domination number   of GG, denoted by γpr(G)γpr(G). In this work, we present several upper bounds on the paired-domination number in terms of the maximum degree, minimum degree, girth and order.