A property of trees in terms of unique connected subgraphs
Stephen T. Hedetniemi, Diane M. Spresser · Journal of Graph Theory · 1977
Abstract The following theorem is proved: Let G be a graph with p ≥ 3 points such that for some n, 3 ≤ n ≤ p, any n points lie on a unique smallest connected subgraph. Then G = Cn+1 or G is a tree, and conversely.