Relating the annihilation number and the total domination number of a tree

A set SS of vertices in a graph GG is a total dominating set if every vertex of GG is adjacent to some vertex in SS. The total domination number γt(G)γt(G) is the minimum cardinality of a total dominating set in GG. The annihilation number a(G)a(G) is the largest integer kk such that the sum of the first kk terms of the non-decreasing degree sequence of GG is at most the number of edges in GG. In this paper, we investigate relationships between the annihilation number and the total domination number of a graph. Let TT be a tree of order n≥2n≥2. We show that γt(T)≤a(T)+1γt(T)≤a(T)+1, and we characterize the extremal trees achieving equality in this bound.