Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T14:26:11.086Z Has data issue: false hasContentIssue false
  • English
  • Français

Reconstruction of Cacti

Published online by Cambridge University Press:  20 November 2018

Dennis Geller  and
Bennet Manvel
Dennis Geller
Affiliation:
The University of Michigan, Ann Arbor, Michigan
Bennet Manvel
Affiliation:
The University of Michigan, Ann Arbor, Michigan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Following the work of Kelly (8), Harary and Palmer (5), and Bondy (1) on the reconstruction of trees, and of Manvel (10) on the reconstruction of connected graphs with a single cycle, it was a natural step to attempt to solve the reconstruction problem for cacti. The solution of this problem, presented here, assumes both Kelly's Theorem and the result of Manvel in (10). Any definitions not given here can be found in (2).

Let graph G have point set V = {v1 v2, …, vp} and line set X = {x1, x2, …, xq}. For each vi ∈ V, Gi = G – vi is the maximal subgraph of G which does not contain vi and is formed by deleting vi and all lines incident with it from G.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1354 - 1360
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Bondy, J. A., Ow Kelly's congruence theorem for trees, Proc. Cambridge Philos. Soc. 65 (1969), 387–397.Google Scholar
2. Harary, F., Graph theory (Addison-Wesley, Reading, Massachusetts, 1969).Google Scholar
3. 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
4. Harary, F. and Anderson, S., Trees and unicyclic graphs, Math. Teacher 60 (1967), 345–348.Google Scholar
5. Harary, F. and Palmer, E. M., The reconstruction of a tree from its maximal subtrees, Can. J. Math. 18 (1966), 803–810.Google Scholar
6. Harary, F. and Uhlenbeck, G. E., On the number of Husimi trees, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 315–322.Google Scholar
7. Husimi, K., Note on Mayer's theory of cluster integrals, J. Chem. Phys. 18 (1950), 682–684.Google Scholar
8. Kelly, P. J., A congruence theorem for trees, Pacific J. Math. 7 (1957), 961–968.Google Scholar
9. Kônig, D., Théorie der endlichen und unendlichen Graphen (Leipzig, Berlin, 1936; reprinted by Chelsea, New York, 1950).Google Scholar
10. Manvel, B., Reconstruction of unicyclic graphs, pp. 103—107 in Proof techniques in graph theory, edited by Harary, F. (Academic Press, New York, 1969).Google Scholar
10. Ulam, S. M., A collection of mathematical problems, p. 29 (Interscience, New York, 1960).Google Scholar