Graphs and Combinatorics

, Volume 32, Issue 6, pp 2525–2539 | Cite as

Computing Planarity in Computable Planar Graphs

  • Oscar Levin
  • Taylor McMillan
Original Paper


A graph is computable if there is an algorithm which decides whether given vertices are adjacent. Having a procedure for deciding the edge set might not help compute other properties or features of the graph, however. The goal of this paper is to investigate the extent to which features related to the planarity of a graph might or might not be computable. We propose three definitions for what it might mean for a computable graph to be computably planar and for each build a computable planar graph which fails to be computably planar. We also consider these definitions in the context of highly computable graphs, those for which there is an algorithm which computes the degree of a given vertex.


Infinite graphs Planar graphs Computability theory Planar embeddings 

Mathematics Subject Classification

05C10 05C63 03D45 



The authors thank the anonymous referee for a number of helpful suggestions.


  1. 1.
    Bean, D.R.: Effective coloration. J. Symbolic Logic 41(2), 469–480 (1976)Google Scholar
  2. 2.
    Bean, D.R.: Recursive Euler and Hamilton paths. Proc. Am. Math. Soc. 55(2), 385–394 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carstens, H.G., Päppinghaus, P.: Recursive coloration of countable graphs. Ann. Pure Appl. Logic 25(1), 19–45 (1983). doi: 10.1016/0168-0072(83)90052-0 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Diestel, R.: Graph theory, Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14279-6
  5. 5.
    Erdös, P.: Some remarks on set theory. Proc. Am. Math. Soc. 1, 127–141 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fáry, I.: On straight line representation of planar graphs. Acta Univ. Szeged. Sect. Sci. Math. 11, 229–233 (1948)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Gasarch, W.: A survey of recursive combinatorics. In: Handbook of Recursive Mathematics, vol. 2, Stud. Logic Found. Math., vol. 139, pp. 1041–1176. North-Holland, Amsterdam (1998). doi: 10.1016/S0049-237X(98)80049-9
  8. 8.
    Harel, D.: Hamiltonian paths in infinite graphs. Israel J. Math. 76(3), 317–336 (1991). doi: 10.1007/BF02773868 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jura, M., Levin, O., Markkanen, T.: Domatic partitions of computable graphs. Arch. Math. Logic 53(1–2), 137–155 (2014). doi: 10.1007/s00153-013-0359-2 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kierstead, H.A.: Recursive colorings of highly recursive graphs. Can. J. Math. 33(6), 1279–1290 (1981). doi: 10.4153/CJM-1981-097-8 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36(2), 177–189 (1979). doi: 10.1137/0136016 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Manaster, A.B., Rosenstein, J.G.: Effective matchmaking (recursion theoretic aspects of a theorem of Philip Hall). Proc. Lond. Math. Soc. 3(25), 615–654 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Schmerl, J.H.: Recursive colorings of graphs. Can. J. Math. 32(4), 821–830 (1980). doi: 10.4153/CJM-1980-062-7 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Simpson, S.G.: Subsystems of second order arithmetic, 2nd edn. Perspectives in Logic. Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie (2009). doi: 10.1017/CBO9780511581007
  15. 15.
    Soare, R.I.: Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. A Study of Computable Functions and Computably Generated Sets. Springer, Berlin (1987). doi: 10.1007/978-3-662-02460-7
  16. 16.
    Thomassen, C.: Straight line representations of infinite planar graphs. J. Lond. Math. Soc. (2) 16(3), 411–423 (1977). doi: 10.1112/jlms/s2-16.3.411 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Northern ColoradoGreeleyUSA

Personalised recommendations