Upper total domination versus upper paired-domination

Let G be a, graph with no isolated vertices. A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S, while a paired-dominating set of G is a dominating set of vertices whose induced subgraph has a perfect matching. The maximum cardinality of a minimal total dominating set and a minimal paired-dominating set of G is the upper total domination number and upper paired-domination number of G, respectively, denoted by Gamma(t)(G) and Gamma(pr)(G). In this paper, we investigate the relationship between the upper total domination and upper paired-domination numbers of a graph. We show that for every graph G with no isolated vertex Gamma(t)(G) >= 1/2 (Gamma(pr) (G) + 2), and we characterize the trees achieving this bound. For each positive integer k., we observe that there exist, connected graphs G and H such that Gamma(pr)(G) - Gamma(t)(G) >= k and Gamma(t)(H) - Gamma(pr)(H) >= k. However for the family of trees T on at least two vertices, we show that Gamma(t)(T) <= Gamma(pr) (T).