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# On the undecidability of finite planar graphs1

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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.

## Footnotes

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1

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

## References

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[1]Blineke, L. W., The decomposition of complete graphs into planar subgraphs. Graph theory and theoretical physics, Academic Press, London, 1967, chapter 4, pp. 139158.
[2]Garfunkel, S., On the undecidability of certain finite theories, Transactions of the American Mathematical Society (to appear).
[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. 5868.
[5]Rabin, M., Decidability of second-order theories and automata of infinite trees, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 135.
[6]Vaught, R., Sentences true in all constructive models, this Journal, vol. 25 (1960), pp. 3953.

# On the undecidability of finite planar graphs1

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