The paired-domination and the upper paired-domination numbers of graphs

In this paper we continue the study of paired-domination in graphs. A paired-dominating set, abbreviated PDS, 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 \(\gamma_{p}(G)\), is the minimum cardinality of a PDS of \(G\). The upper paired-domination number of \(G\), denoted by \(\Gamma_{p}(G)\), is the maximum cardinality of a minimal PDS of \(G\). Let \(G\) be a connected graph of order \(n\geq 3\). Haynes and Slater in [Paired-domination in graphs, Networks 32 (1998), 199-206], showed that \(\gamma_{p}(G)\leq n-1\) and they determine the extremal graphs \(G\) achieving this bound. In this paper we obtain analogous results for \(\Gamma_{p}(G)\). Dorbec, Henning and McCoy in [Upper total domination versus upper paired-domination, Questiones Mathematicae 30 (2007), 1-12] determine \(\Gamma_{p}(P_n)\), instead in this paper we determine \(\Gamma_{p}(C_n)\). Moreover, we describe some families of graphs \(G\) for which the equality \(\gamma_{p}(G)=\Gamma_{p}(G)\) holds