On the Minimum Graphs Which Contain all Small Trees

Abstract

Let $E_n$ denote the minimum number of edges in a graph that contains every tree with $n$ edges. This article provides two sets of data concerning $(n+1)$-vertex graphs with $E_n$ edges for each $n\le 11$: first, a minimum set of trees with $n$ edges such that all trees with $n$ edges are contained in such a graph whenever it contains the trees in the minimum set; second, all mutually nonisomorphic graphs that contain all trees with $n$ edges.