Saturation Numbers for Trees

For a fixed graph H, a graph G is H-saturated if there is no copy of H in G, but for any edge e 6∈ G, there is a copy of H in G + e. The collection of Hsaturated graphs of order n is denoted by SAT(n,H), and the saturation number, sat(n,H), is the minimum number of edges in a graph in SAT(n,H). Let Tk be a tree on k vertices. The saturation numbers sat(n, Tk) for some families of trees will be determined precisely. Some classes of trees for which sat(n, Tk) < n will be identified, and trees Tk in which graphs in SAT(n, Tk) are forests will be presented. Also, families of trees for which sat(n, Tk) ≥ n will be presented. The maximum and minimum values of sat(n, Tk) for the class of all trees will be given. Some properties of sat(n, Tk) and SAT(n, Tk) for trees will be discussed.