Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T18:04:43.815Z Has data issue: false hasContentIssue false
  • English
  • Français

On the undecidability of finite planar graphs1

Published online by Cambridge University Press:  12 March 2014

Solomon Garfunkel  and
Herbert Shank
Solomon Garfunkel
Affiliation:
Cornell University, Ithaca, New York
Herbert Shank
Affiliation:
Cornell University, Ithaca, New York
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we demonstrate the hereditary undecidability of finite planar graphs. In §2 we introduce the preliminary logical notions used and outline the Rabin–Scott method of semantic embedding. This method is illustrated in §3 by proving the undecidability of the theory of two finite equivalence relations of a special type. In §4 we give a proof of the main theorem by embedding these equivalence relations into finite planar graphs.

The basic idea is first to form a graph which codes a pair of these relations and then to take a representative of it and “squish” it to the plane. This “squishing” requires the introduction of crossings; and edges of the original graph become paths in the new one. To distinguish the original edges we place two different types of “diamonds” about crossing points. We can then uncode our new graphs to recover the equivalence relations by means of simple first-order incidence properties.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 1 , March 1971 , pp. 121 - 126
Copyright
Copyright © Association for Symbolic Logic 1971

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

1

This research was supported in part by National Science Foundation Grant GP-8732 and AF-AFOSR-68–1402.

References

[1]Blineke, L. W., The decomposition of complete graphs into planar subgraphs. Graph theory and theoretical physics, Academic Press, London, 1967, chapter 4, pp. 139–158.Google Scholar
[2]Garfunkel, S., On the undecidability of certain finite theories, Transactions of the American Mathematical Society (to appear).Google Scholar
[3]Harary, F., Graph theory, Addison Wesley, Reading, Mass., 1969.CrossRefGoogle Scholar
[4]Rabin, M., A simple method for undecidability proofs, Proceedings of the 1964 International Congress for Logic, Methodology and Philosophy of Science, Jerusalem, 1964. North-Holland, Amsterdam, pp. 58–68.Google Scholar
[5]Rabin, M., Decidability of second-order theories and automata of infinite trees, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 1–35.Google Scholar
[6]Vaught, R., Sentences true in all constructive models, this Journal, vol. 25 (1960), pp. 39–53.Google Scholar