Reconstruction of Trees

Abstract

Every tree T determines a set of distinct maximal proper subtrees T i = T — v i , which are obtained by the deletion of an endpoint of T . In this paper we prove that a tree is almost always uniquely determined by this set of its subtrees, and point out two interesting consequences of this result. In [ 5 ], Ulam proposed the following conjecture, which we state in a slightly stronger form due to Harary [ 1 ]. ULAM'S CONJECTURE. A graph G with at least three points is uniquely determined up to isomorphism by the subgraphs G i = G — v i . Kelly [ 4 ] proved the conjecture for trees and Harary and Palmer [ 3 ] showed that not all of the G i are needed in that case by proving Corollary 1 below. If we remove from the list of subgraphs G i of a graph G all but one graph of each isomorphism type, we obtain a set of G i which are distinct up to isomorphism.