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Reconstruction of Trees

  • Bennet Manvel (a1)

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Every tree T determines a set of distinct maximal proper subtrees Ti = Tvi , 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 = Gvi .

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.

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References

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1. Harary, F., On the reconstruction of a graph from a collection of subgraphs, pp. 4752 in Theory of graphs and its applications, edited by Fiedler, M. (Academic Press, New York, 1964).
2. Harary, F., Graph theory (Addison-Wesley, Reading, Massachusetts 1969).
3. Harary, F. and Palmer, E. M., The reconstruction of a tree from its maximal subtrees, Can. J. Math. 18 (1966), 803810.
4. Kelly, P. J., A congruence theorem for trees, Pacific J. Math. 7 (1957), 961968.
5. Ulam, S. M., A collection of mathematical problems, p. 29 (Wiley (Interscience), New York, 1960).
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Reconstruction of Trees

  • Bennet Manvel (a1)

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