Mathematical systems theory

, Volume 27, Issue 3, pp 257–273 | Cite as

On the complexity of graph reconstruction

  • Dieter Kratsch
  • Lane A. Hemaspaandra
Article
Received:
Revised:

Abstract

In the wake of the resolution of the four-color conjecture, the graph reconstruction conjecture has emerged as one focal point of graph theory. This paper considers thecomputational complexity of decision problems (Deck Checking andLegitimate Deck), construction problems (Preimage Construction), and counting problems (Preimage Counting) related to the graph reconstruction conjecture. We show that:
$${\text{GRAPH}} {\text{ISOMORPHISM}} \leqslant _m^l LEGITIMATE DECK, and$$
(1.)
$${\text{GRAPH}} {\text{ISOMORPHISM}} \equiv _{iso}^l DECK CHECKING.$$
(2)

Relatedly, we display the first natural GI-hard NP set lacking obvious padding functions. Finally, we show thatLegitimate Deck, Preimage Construction, andPreimage Counting are solvable in polynomial time for graphs of bounded degree, partialk-trees for any fixedk, and graphs of bounded genus, in particular for planar graphs.

Keywords

Graph Theory Computational Mathematic Polynomial Time Focal Point Decision Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Be]
    C. Berge.Graphs. North-Holland, Amsterdam, 1985.Google Scholar
  2. [BGM]
    L. Babai, D. Grigoryev, and D. Mount Isomorphism of graphs with bounded eigenvalue multiplicity.Proceedings of the 14th ACM Symposium on Theory of Computing, pages 310–324, 1982.Google Scholar
  3. [BH1]
    L. Berman and J. Hartmanis. On isomorphisms and density of NP and other complete sets.SIAM J. Comput., 6(2):305–322, 1977.Google Scholar
  4. [BH2]
    J. Bondy and R. Hemminger. Graph reconstruction—a survey.J. Graph Theory, 1:227–268. 1977.Google Scholar
  5. [BHZ]
    R. Boppana, J. Hastad, and S. Zachos. Does co-NP have short interactive proofs?Inform. Process. Lett., 25:127–132, 1987.Google Scholar
  6. [Bod1]
    H. Bodlaender. Polynomial Algorithms for Chromatic Index and Graph Isomorphism on Partial k-trees.J. Algorithms, 11:631–643, 1990.Google Scholar
  7. [Bod2]
    H. Bodlaender. Dynamic programming of graphs with bounded treewidth.Proceedings of the 15th International Colloquium on Automata, Languages, and Programming, pages 105–119. Lecture Notes in Computer Science, Vol. 317. Springer-Verlag, Berlin, 1988.Google Scholar
  8. [Bon]
    J. Bondy. On Ulam's conjecture for separable graphs.Pacific J. Math., 31:281–288, 1969.Google Scholar
  9. [Boo]
    K. Booth. Isomorphism testing for graphs, semigroups, and finite automata are polynomially equivalent problems.SIAM J. Comput., 7(3):273–279, 1978.Google Scholar
  10. [Ch]
    P. Chinn. A graph with ϱ points and enough distinct (ϱ — 2)-order subgraphs is reconstructible. In M. Capobianco, J. B. Frechen, and M. Krolik, editors,Recent Trends in Graph Theory, pages 71–73. Lecture Notes in Mathematics, Vol. 186. Springer-Verlag, Berlin, 1971.Google Scholar
  11. [CKS]
    G. Chartrand, V. Kronk, and S. Schuster. A technique for reconstructing disconnected graphs.Colloq. Math., 27:31–34, 1973.Google Scholar
  12. [Di]
    P. Dietz. Intersection Graph Algorithms. Ph.D. thesis. Computer Science Department, Cornell University, Ithaca, NY, 1984.Google Scholar
  13. [Do]
    W. Doerfler. Some results on the reconstruction of graphs.Infinite and Finite Sets, pages 361–383. Colloquia Mathematica Societatis Janos Bolayi, Vol. 10. North-Holland, Amsterdam, 1975.Google Scholar
  14. [FFK]
    S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes.Proceedings of the 6th Structure in Complexity Theory Conference, pages 30–42. IEEE Computer Society Press, New York, 1991.Google Scholar
  15. [FM]
    I. Filotti and J. Mayer. A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus.Proceedings of the 12th ACM Symposium on Theory of Computing, pages 236–243, April 1980.Google Scholar
  16. [GH]
    D. Greenwell and R. Hemminger. Reconstructing graphs. In G. Chartrand and S. F. Kapoor, editors,The Many Facets of Graph Theory, pages 91–114. Lecture Notes in Mathematics, Vol. 110. Springer-Verlag, Berlin, 1969.Google Scholar
  17. [Gil]
    W. Giles. The reconstruction of outerplanar graphs.J. Combin. Theory Ser. B, 16:215–226, 1974.Google Scholar
  18. [Gi2]
    J. Gill. Computational complexity of probabilistic Turing machines.SIAM J. Comput., 6(4):675–695, 1977.Google Scholar
  19. [GJ]
    M. Garey and D. Johnson.Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979.Google Scholar
  20. [GM]
    D. Geller and B. Manvel. Reconstruction of cacti.Canad. J. Math., 21:1354–1360, 1969.Google Scholar
  21. [GMW]
    O. Goldreich, S. Micali, and A. Widgerson. Proofs that yield nothing but their validity and a methodology of cryptographic protocol design.Proceedings of the 27th IEEE Symposium on Foundations of Computer Science, pages 174–187, April 1986.Google Scholar
  22. [Go]
    M. Golumbic.Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.Google Scholar
  23. [GS]
    S. Goldwasser and M. Sipser. Private coins versus public coins in interactive proof systems.Proceedings of the 18th ACM Symposium on Theory of Computing, pages 59–68, 1986.Google Scholar
  24. [GT]
    J. Gross and T. Tucker.Topological Graph Theory. Wiley, New York, 1987.Google Scholar
  25. [Gu]
    S. Gupta. Reconstruction conjecture for square of a tree. In K. M. Koh and H. P. Yap, editors,Graph Theory, pages 268–278. Lecture Notes in Mathematics, Vol. 1073. Springer-Verlag, Berlin, 1984.Google Scholar
  26. [Hal]
    F. Harary. On the reconstruction of a graph from a collection of subgraphs. In M. Fiedler, editor,Theory of Graphs and Its Applications, pages 47–52. Prague, 1964.Google Scholar
  27. [Ha2]
    F. Harary.Graph Theory. Addison-Wesley, Reading, MA, 1969.Google Scholar
  28. [Ha3]
    F. Harary. A survey of the reconstruction conjecture. In R. A. Bari and F. Harary, editors,Graphs and Combinatorics, pages 18–28. Lecture Notes in Mathematics, Vol. 406. Springer-Verlag, Berlin, 1974.Google Scholar
  29. [Ha4]
    J. Hartmanis. On log-tape isomorphisms of complete sets.Theoret. Comput. Sci., 7:273–286, 1978.Google Scholar
  30. [HK]
    J. Hopcroft and R. Karp. Ann 5/2 algorithm for maximum matching in bipartite graphs.SIAM J. Comput., 2:225–231, 1973.Google Scholar
  31. [HU]
    J. Hopcroft and J. Ullman.Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading, MA, 1979.Google Scholar
  32. [Jol]
    D. Johnson. The NP-completeness column: an ongoing guide.J. Algorithms, 6:434–451, 1985.Google Scholar
  33. [Jo2]
    D. Johnson. A catalog of complexity classes. In J. Van Leeuwen, editor,Handbook of Theoretical Computer Science, Chapter 2, pages 67–161. MIT Press/Elsevier, Cambridge, MA/Amsterdam, 1990.Google Scholar
  34. [JY]
    D. Joseph and P. Young. Some remarks on witness functions for non-polynomial and non-complete sets in NP.Theoret. Comput. Sci., 39:225–237, 1985.Google Scholar
  35. [Ke]
    P. Kelly. A congruence theorem for trees.Pacific J. Math., 7:961–968, 1957.Google Scholar
  36. [KST1]
    K. Köbler, U. Schöning, and T. Thierauf. Personal communication, 1991.Google Scholar
  37. [KST2]
    J. Köbler, U. Schöning, and J. Torán. Graph isomorphism is low for PP.Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science, pages 401–411. Lecture Notes in Computer Science, Vol. 577. Springer-Verlag, Berlin, 1992.Google Scholar
  38. [Lal]
    J. Lauri. Proof of Harary's conjecture on reconstruction of trees.Discrete Math., 43:79–90, 1983.Google Scholar
  39. [La2]
    J. Lauri. Graph reconstruction—some techniques and new problems.Ars Combin., 24(B):35–61, 1987.Google Scholar
  40. [LL]
    R. Ladner and N. Lynch. Relativization of questions about log space computability.Math. Systems Theory, 10(l):19–32, 1976.Google Scholar
  41. [LLS]
    R. Ladner, N. Lynch, and A. Selman. A comparison of polynomial time reducibilities.Theoret. Comput. Sci., 1(2):103–124, 1975.Google Scholar
  42. [Li]
    T. Long. Strong nondeterministic polynomial-time reducibilities.Theoret. Comput. Sci., 21:1–25, 1982.Google Scholar
  43. [Lu]
    E. Luks. Isomorphism of graphs of bounded valence can be tested in polynomial time.J. Comput. System Sci., 25:42–65, 1982.Google Scholar
  44. [Mal]
    B. Manvel. Reconstruction of unicyclic graphs. In F. Harary, editor,Proof Techniques in Graph Theory, pages 103–107. Academic Press, New York, 1969.Google Scholar
  45. [Ma2]
    B. Manvel. Reconstruction of maximal outerplanar graphs.Discrete Math., 2:269–278, 1972.Google Scholar
  46. [Mi]
    G. Miller. Isomorphism testing for graphs of bounded genus.Proceedings of the 12th ACM Symposium on Theory of Computing, pages 225–235, 1980.Google Scholar
  47. [MV]
    S. Micali and V. Vazirani. AnO(V1/2E) algorithm for finding maximum matching in general graphs.Proceedings of the 21st Annual Symposium on Foundations of Computer Science, pages 17–27, New York, 1980.Google Scholar
  48. [MW]
    B. Manvel and J. Weinstein. Nearly acyclic graphs are reconstructible.J Graph Theory, 2:25–39, 1978.Google Scholar
  49. [MY]
    S. Mahaney and P. Young. Reductions among polynomial isomorphism types.Theoret. Comput. Sci., 39:207–224, 1985.Google Scholar
  50. [NW]
    C. St. J. A. Nash-Williams. The reconstruction problem. In L. W. Beineke and R. J. Wilson, editors,Selected Topics in Graph Theory, pages 205–236. Academic Press, New York, 1978.Google Scholar
  51. [Ri]
    M. von Rimscha. Reconstructibility and perfect graphs.Discrete Math., 47:79–90, 1983.Google Scholar
  52. [RS]
    N. Robertson and P. D. Seymour. Graph minors—a survey. In I. Anderson, editor,Surveys in Combinatorics 1985: Invited Papers for the Tenth British Combinatorial Conference, pages 153–171. Cambridge University Press, Cambridge, 1985.Google Scholar
  53. [Sc1]
    U. Schöning. A low and high hierarchy in NP.J. Comput. System Sci., 27:14–28, 1983.Google Scholar
  54. [Sc2]
    U. Schöning. Graph isomorphism is in the low hierarchy.J. Comput. System Sci., 37:312–323, 1988.Google Scholar
  55. [Se]
    A. Selman. Polynomial time enumeration reducibility.SIAM J. Comput., 7(4):440–457, 1978.Google Scholar
  56. [Si]
    J. Simon. On Some Central Problems in Computational Complexity. Ph.D. thesis, Cornell University, Ithaca, NY, January 1975. Available as Cornell Department of Computer Science Technical Report TR75-224.Google Scholar
  57. [Ul]
    S. Ulam.A Collection of Mathematical Problems. Interscience, New York, 1960.Google Scholar
  58. [Wa]
    K. Wagner. The complexity of combinatorial problems with succinct input representations.Acta Inform., 23:325–356, 1986.Google Scholar
  59. [WB]
    A. White and L. Beineke. Topological graph theory. In L. W. Beineke and R. J. Wilson, editors,Selected Topics in Graph Theory, pages 15–49. Academic Press, New York, 1978.Google Scholar
  60. [WW]
    K. Wagner and G. Wechsung.Computational Complexity. Mathematics and Its Applications. Reidel, Dordrecht, 1986.Google Scholar
  61. [Yo]
    P. Young. Juris Hartmanis: fundamental contributions to isomorphism problems. In A. Selman, editor,Complexity Theory Retrospective, pages 28–58. Springer-Verlag, New York, 1990.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1994

Authors and Affiliations

  • Dieter Kratsch
    • 1
  • Lane A. Hemaspaandra
    • 2
  1. 1.Fakultät für Mathematik und InformatikFriedrich-Schiller-UniversitätJenaGermany
  2. 2.Department of Computer ScienceUniversity of RochesterRochesterUSA

Personalised recommendations