Graphs with large paired-domination number

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (1998) Networks 32: 199-206. A paired-dominating set of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by γ_pr(G), is the minimum cardinality of a paired-dominating set of G. Let G be a connected graph of order n with minimum degree at least two. Haynes and Slater (1998) Networks 32: 199-206, showed that if n ≥ 6, then γ_pr(G) ≤ 2n/3. In this paper, we show that there are exactly ten graphs that achieve equality in this bound. For n ≥ 14, we show that γ_pr(G)≤ 2(n-1)/3 and we characterize the (infinite family of) graphs that achieve equality in this bound.