# Definability of Recursive Predicates in the Induced Subgraph Order

Conference paper

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## Abstract

Consider the set of all finite simple graphs \(\mathcal {G}\) ordered by the induced subgraph order \(\le _i\). Building on previous work by Wires [14] and Jezek and Mckenzie [5, 6, 7, 8], we show that every recursive predicate over graphs is definable in the first order theory of (\(\mathcal {G},\le _i, P_3\)) where \(P_3\) is the path on 3 vertices.

### Keywords

Sugar## Notes

### Acknowledgment

I would like to thank my guide Prof. R. Ramanujam for his advice and discussions on both technical matter and the presentation of this paper.

### References

- 1.Cook, S., Nguyen, P.: Logical Foundations of Proof Complexity. Cambridge University Press, Cambridge (2010)CrossRefMATHGoogle Scholar
- 2.Enderton, H.: A Mathematical Introduction to logic. Academic Press, Burlington (2001)MATHGoogle Scholar
- 3.Fitting, M.: Fundamentals of Generalized Recursion Theory. Elsevier, Amsterdam (2011)MATHGoogle Scholar
- 4.Grohe, M.: The quest for a logic capturing PTIME. In: 23rd Annual IEEE Symposium on Logic in Computer Science, LICS 2008, pp. 267–271. IEEE (2008)Google Scholar
- 5.Ježek, J., McKenzie, R.: Definability in substructure orderings, IV: finite lattices. Algebra Univers.
**61**(3–4), 301–312 (2009)MathSciNetMATHGoogle Scholar - 6.Ježek, J., McKenzie, R.: Definability in substructure orderings, I: finite semilattices. Algebra Univers.
**61**(1), 59–75 (2009)MathSciNetCrossRefMATHGoogle Scholar - 7.Ježek, J., McKenzie, R.: Definability in substructure orderings, III: finite distributive lattices. Algebra Univers.
**61**(3–4), 283–300 (2009)MathSciNetMATHGoogle Scholar - 8.Ježek, J., McKenzie, R.: Definability in substructure orderings, II: finite ordered sets. Order
**27**(2), 115–145 (2010)MathSciNetCrossRefMATHGoogle Scholar - 9.Kaye, R.: Models of Peano arithmetic. Oxford University Press, Oxford (1991)MATHGoogle Scholar
- 10.Krajicek, J.: Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, Cambridge (1995)CrossRefMATHGoogle Scholar
- 11.Kunos, Á.: Definability in the embeddability ordering of finite directed graphs. Order
**32**(1), 117–133 (2015)MathSciNetCrossRefMATHGoogle Scholar - 12.Kuske, D.: Theories of orders on the set of words. RAIRO Theor. Inform. Appl.
**40**(01), 53–74 (2006)MathSciNetCrossRefMATHGoogle Scholar - 13.Ramanujam, R., Thinniyam, R.S.: Definability in first order theories of graph orderings. In: Artemov, S., Nerode, A. (eds.) LFCS 2016. LNCS, vol. 9537, pp. 331–348. Springer, Heidelberg (2016). doi: 10.1007/978-3-319-27683-0_23 CrossRefGoogle Scholar
- 14.Wires, A.: Definability in the substructure ordering of simple graphs. Ann. Comb.
**20**(1), 139–176 (2016)MathSciNetCrossRefMATHGoogle Scholar

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