O FAVARON - Department of Mathematics, Univ Paris-Sub, LRI, UMR ۸۶۲۳, Orsay, F-۹۱۴۰۵, France.
H KARAMI - Faculty of Mathematics, Sharif University of Technology, Tehran, Iran
S.M SHAIKHOLESLAM - Department of Mathematics, Azarbayejan University of Tarbiat Moallem, Azarshahr, Tabriz, Iran

A paired-dominating set of a graph G=(V,E) with no isolated vertex is a dominating set of vertices inducing a graph with a perfect matching. The paired-domination number of G, denoted by ypr(G), is the minimum caredinality of a paired-dominating set of G. We consider graphs of order n≥6, minimum degree δ such that G and G' do not have isolated vertex and we shall prove that - if ypr (G) > 4 and ypr(G') > 4, then ypr(G) + ypr(G') ≤ 3+ min {δ(G), δ(G')}. - if δ(G) ≥ 2 and δ(G') ≥ 2, then ypr(G) + ypr(G') ≤ 2n/3 + 4 and ypr(G) + ypr(G') ≤ 2n/3 + 2 if moreover n ≥ 21.

paired-domination number, Nordhaus- Guddum inequalities