Abstract
Every tree T determines a set of distinct maximal proper subtrees Ti = T — vi, 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 Gi = G — vi.Kelly [4] proved the conjecture for trees and Harary and Palmer [3] showed that not all of the Gi are needed in that case by proving Corollary 1 below. If we remove from the list of subgraphs Gi of a graph G all but one graph of each isomorphism type, we obtain a set of Gi which are distinct up to isomorphism.
Publisher
Canadian Mathematical Society
Cited by
26 articles.
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2. On some properties of trees with reduced deck of size 1;Proceedings of Academician O.B. Lupanov 14th International Scientific Seminar "Discrete Mathematics and Its Applications";2022
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4. Reconstruction of Rooted Directed Trees;Acta Cybernetica;2019-11-03
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