Skip to main content
SpringerLink
Log in

Comparing Universal Covers in Polynomial Time

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

The universal cover T G of a connected graph G is the unique (possibly infinite) tree covering G, i.e., that allows a locally bijective homomorphism from T G to G. It is well-known that if a graph G covers a graph H, then their universal covers are isomorphic, and that the latter can be tested in polynomial time by checking if G and H share the same degree refinement matrix. We extend this result to locally injective and locally surjective homomorphisms by following a very different approach. Using linear programming techniques we design two polynomial time algorithms that check if there exists a locally injective or a locally surjective homomorphism, respectively, from a universal cover T G to a universal cover T H (both given by their degree matrices). This way we obtain two heuristics for testing the corresponding locally constrained graph homomorphisms. Our algorithm can also be used for testing (subgraph) isomorphism between universal covers, and for checking if there exists a locally injective or locally surjective homomorphism (role assignment) from a given tree to an arbitrary graph H.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abello, J., Fellows, M.R., Stillwell, J.C.: On the complexity and combinatorics of covering finite complexes. Aust. J. Comb. 4, 103–112 (1991)

    MATH  MathSciNet  Google Scholar 

  2. Angluin, D.: Local and global properties in networks of processors. In: Proceedings of the 12th ACM Symposium on Theory of Computing, pp. 82–93 (1980)

  3. Angluin, D., Gardiner, A.: Finite common coverings of pairs of regular graphs. J. Comb. Theory B 30, 184–187 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  4. Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1974)

    MATH  Google Scholar 

  5. Biggs, N.: Constructing 5-arc transitive cubic graphs. J. Lond. Math. Soc. II 26, 193–200 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bodlaender, H.L.: The classification of coverings of processor networks. J. Parallel Distrib. Comput. 6, 166–182 (1989)

    Article  Google Scholar 

  7. Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybern. 11(1–2), 1–22 (1993)

    MATH  MathSciNet  Google Scholar 

  8. Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. American Elsevier, New York (1976)

    Google Scholar 

  9. Chalopin, J., Métivier, Y., Zielonka, W.: Local computations in graphs: the case of cellular edge local computations. Fundam. Inform. 74(1), 85–114 (2006)

    MATH  Google Scholar 

  10. Dantchev, S., Martin, B.D., Stewart, I.A.: On non-definability of unsatisfiability. Manuscript

  11. Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19, 248–264 (1972)

    Article  MATH  Google Scholar 

  12. Everett, M.G., Borgatti, S.: Role colouring a graph. Math. Soc. Sci. 21(2), 183–188 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fiala, J., Kratochvíl, J.: Complexity of partial covers of graphs. In: Eades, P., Takaoka, T. (eds.) ISAAC. Lecture Notes in Computer Science, vol. 2223, pp. 537–549. Springer, New York (2001)

    Google Scholar 

  14. Fiala, J., Kratochvíl, J.: Partial covers of graphs. Discuss. Math. Graph Theory 22, 89–99 (2002)

    MATH  MathSciNet  Google Scholar 

  15. Fiala, J., Maxová, J.: Cantor-Bernstein type theorem for locally constrained graph homomorphisms. Eur. J. Comb. 7(27), 1111–1116 (2006)

    Article  Google Scholar 

  16. Fiala, J., Paulusma, D.: A complete complexity classification of the role assignment problem. Theor. Comput. Sci. 1(349), 67–81 (2005)

    Article  MathSciNet  Google Scholar 

  17. Fiala, J., Paulusma, D.: Comparing universal covers in polynomial time. In: Hirsch, E.A., Razborov, A.A., Semenov, A.L., Slissenko, A. (eds.) CSR. Lecture Notes in Computer Science, vol. 5010, pp. 158–167. Springer, Berlin (2008)

    Google Scholar 

  18. Fiala, J., Kratochvíl, J., Kloks, T.: Fixed-parameter complexity of λ-labelings. Discrete Appl. Math. 113(1), 59–72 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fiala, J., Heggernes, P., Kristiansen, P., Telle, J.A.: Generalized H-coloring and H-covering of trees. Nord. J. Comput. 10(3), 206–224 (2003)

    MATH  MathSciNet  Google Scholar 

  20. Fiala, J., Golovach, P.A., Kratochvíl, J.: Distance constrained labelings of graphs of bounded treewidth. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP. Lecture Notes in Computer Science, vol. 3580, pp. 360–372. Springer, New York (2005)

    Google Scholar 

  21. Fiala, J., Paulusma, D., Telle, J.A.: Locally constrained graph homomorphism and equitable partitions. Eur. J. Comb. 29(4), 850–880 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Godsil, C.D.: Algebraic Combinatorics. Chapman and Hall, New York (1993)

    MATH  Google Scholar 

  23. Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press, Oxford (2004)

    Book  MATH  Google Scholar 

  24. Hoffman, A.J., Kruskal, J.G.: Integral boundary points of convex polyhedra. Ann. Math. Stud. 38, 223–246 (1956)

    MATH  MathSciNet  Google Scholar 

  25. Kranakis, E., Krizanc, D., van den Berg, J.: Computing Boolean functions on anonymous networks. Inf. Comput. 114(2), 214–236 (1994)

    Article  MATH  Google Scholar 

  26. Kratochvíl, J., Proskurowski, A., Telle, J.A.: Covering directed multigraphs I. Colored directed multigraphs. In: Möhring, R.H. (ed.) WG. Lecture Notes in Computer Science, vol. 1335, pp. 242–257. Springer, Berlin (1997)

    Google Scholar 

  27. Kratochvíl, J., Proskurowski, A., Telle, J.A.: Covering regular graphs. J. Comb. Theory, Ser. B 71(1), 1–16 (1997)

    Article  MATH  Google Scholar 

  28. Kristiansen, P., Telle, J.A.: Generalized H-coloring of graphs. In: Lee, D.T., Teng, S.-H. (eds.) ISAAC. Lecture Notes in Computer Science, vol. 1969, pp. 456–466. Springer, Berlin (2000)

    Google Scholar 

  29. Leighton, F.T.: Finite common coverings of graphs. J. Comb. Theory B 33, 231–238 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  30. Massey, W.S.: Algebraic Topology: An Introduction. Harcourt, Brace and World, New York (1967)

    MATH  Google Scholar 

  31. Moore, E.F.: Gedanken-experiments on sequential machines. In: Automata Studies. Annals of Mathematics Studies, vol. 34, pp. 129–153. Princeton University Press, Princeton (1956)

    Google Scholar 

  32. Nešetřil, J.: Homomorphisms of derivative graphs. Discrete Math. 1(3), 257–268 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  33. Norris, N.: Universal covers of graphs: isomorphism to depth n−1 implies isomorphism to all depths. Discrete Appl. Math. 56(1), 61–74 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  34. Pekeč, A., Roberts, F.S.: The role assignment model nearly fits most social networks. Math. Soc. Sci. 41(3), 275–293 (2001)

    Article  MATH  Google Scholar 

  35. Reidemeister, K.: Einführung in die kombinatorische Topologie. Braunschweig: Friedr. Vieweg & Sohn A.-G. XII, 209 S. (1932)

  36. Roberts, F.S., Sheng, L.: How hard is it to determine if a graph has a 2-role assignment? Networks 37(2), 67–73 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  37. Schwenk, A.J.: Computing the characteristic polynomial of a graph. In: Graphs and Combinatorics. Lecture Notes in Mathematics, vol. 406, pp. 153–172. Springer, Berlin (1974)

    Chapter  Google Scholar 

  38. Shamir, R., Tsur, D.: Faster subtree isomorphism. J. Algorithms 33(2), 267–280 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  39. Yamashita, M., Kameda, T.: Computing on anonymous networks: Part I—characterizing the solvable cases. IEEE Trans. Parallel Distrib. Syst. 7(1), 69–89 (1996)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Physics, DIMATIA and Institute for Theoretical Computer Science (ITI), Charles University, Malostranské nám. 2/25, 118 00, Prague, Czech Republic

    Jiří Fiala

  2. Science Laboratories, Department of Computer Science, Durham University, South Road, Durham, DH1 3EY, England

    Daniël Paulusma

Authors
  1. Jiří Fiala
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniël Paulusma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiří Fiala.

Additional information

A preliminary version of this article has been published in proceedings of the 3rd International Computer Science Symposium in Russia [17].

ITI supported by the Ministry of Education of the Czech Republic as project 1M0021620808.

D. Paulusma supported by EPSRC as project EP/D053633/1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fiala, J., Paulusma, D. Comparing Universal Covers in Polynomial Time. Theory Comput Syst 46, 620–635 (2010). https://doi.org/10.1007/s00224-009-9200-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-009-9200-z

Keywords

Search

Navigation