Graphs with large paired-domination number

Abstract

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 $\gamma_{\rm pr}(G)$, 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 $\gamma_{\rm pr}(G) \le 2n/3$. In this paper, we show that there are exactly ten graphs that achieve equality in this bound. For n ≥ 14, we show that $\gamma_{\rm pr}(G) \le 2(n-1)/3$ and we characterize the (infinite family of) graphs that achieve equality in this bound.