An improved upper bound on the domination number of a tree

A set D of vertices in a graph G is a dominating set of G if every vertex in V(G) \ D is adjacent to at least one vertex in D. The domination number of G, denoted by gamma(G), is the minimum cardinality among all dominating sets of G. In this article, we first provide an alternative proof for the bound gamma(T) <= (n(T)+ |S(T)|)/3 obtained recently by Cabrera -Martinez et al. (2023). Moreover, we improve this previous upper bound. In particular, we show that if T is a tree of order n(T) > 2, then n(T)+ |S(T)| - |SL(T)| |Ls(T)| - |Ss(T)| gamma(T) <= - , 3 3 where S(T), SL(T), Ls(T) and Ss(T) are the sets of support vertices, support link vertices, strong leaves and strong support vertices of T, respectively. Finally, we characterize the trees attaining each of these upper bounds above. (c) 2023 Elsevier B.V. All rights reserved.