Total and Paired-domination Numbers of a Tree

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S . A paired-dominating set of G is a dominating set whose induced subgraph has a perfect matching. The minimum cardinality of a total dominating set (respectively, a paired-dominating set) is the total domination number γt(G) (respectively, the paired-domination number γpr(G) ). We give sharp upper bounds on the total and paired-domination numbers of trees that improve known bounds for some cases. In particular, we show that for a tree T with order n ≥ 3 and s support vertices, γpr(T ) 6 γt(T ) + s − 1 , γt(T ) ≤ (n+ s)/2 , and γpr(T ) ≤ (n+ 2s− 1)/2 .