Saturation Numbers for Trees

Abstract

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