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Reconstruction of Trees

Published online by Cambridge University Press:  20 November 2018

Bennet Manvel
Bennet Manvel*
Affiliation:
The University of Michigan, Ann Arbor, Michigan
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 55 - 60
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Harary, F., On the reconstruction of a graph from a collection of subgraphs, pp. 47–52 in Theory of graphs and its applications, edited by Fiedler, M. (Academic Press, New York, 1964).Google Scholar
2. Harary, F., Graph theory (Addison-Wesley, Reading, Massachusetts 1969).Google Scholar
3. Harary, F. and Palmer, E. M., The reconstruction of a tree from its maximal subtrees, Can. J. Math. 18 (1966), 803–810.Google Scholar
4. Kelly, P. J., A congruence theorem for trees, Pacific J. Math. 7 (1957), 961–968.Google Scholar
5. Ulam, S. M., A collection of mathematical problems, p. 29 (Wiley (Interscience), New York, 1960).Google Scholar