Advertisement

On the equivalence in complexity among basic problems on bipartite and parity graphs

  • Serafino Cicerone
  • Gabriele Di Stefano
Session 7B
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1350)

Abstract

Parity graphs form a superclass of bipartite graphs. Since their introduction, all the algorithms proposed as solutions to the recognition problem and other computational problems exploit the structural property described by Burlet and Uhry in [4].

This paper newly describes a different structural property based on split decomposition for parity graphs. This result, together with the observation that the split decomposition process can be performed in linear time, allows us to provide optimum algorithms for both the recognition problem and the maximum weighted clique problem in this class. We further propose a general algorithm to solve the maximum weighted independent set problem for certain classes of graphs. Its application to parity graphs provides the best solution for this problem. Moreover, when the algorithm is applied to distance-hereditary graphs, it equals the best known solution, which is also the optimum for this class.

A remarkable consequence of this work is that the extension of bipartite graphs to parity graphs does not increase the complexity of these basic problems, since the worst case occurs when the parity graph is an undecomposable bipartite graph.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. S. Adhar and S. Peng. Parallel algorithms for cographs and parity graphs with applications. Journal of Algorithms, 11(2):252–284, June 1990.Google Scholar
  2. 2.
    H. J. Bandelt and M. Mulder. Metric characterization of parity graphs. DMATH: Discrete Mathematics, 91:221–230, 1991.Google Scholar
  3. 3.
    A. Bouchet. Transforming trees by successive local complementations. Journal of Graph Theory, 4:196–207, 1988.Google Scholar
  4. 4.
    M. Burlet and J. P. Uhry. Parity graphs. In C. Berge and V. Chvital, editors, Topics on Perfect Graphs, number 21 in Annals of Discrete Mathematics, pages 253–277. North-Holland, 1984.Google Scholar
  5. 5.
    S. Cicerone and G. Di Stefano. Graph classes between parity and distance-hereditary graphs. In DMTCS '96: Discrete Mathematics and Theoretical Computer Science, Combinatorics, Complexity, and Logic, pages 168–181. Springer-Verlag, 1996.Google Scholar
  6. 6.
    S. Cicerone and G. Di Stefano. Linear time algorithms for parity graphs. Technical Report R.96-16, Dipaxtimento di Ingegneria Elettrica, University di L'Aquila (L'Aquila, Italy), 1996. Submitted for pubblication.Google Scholar
  7. 7.
    S. Cicerone and G. Di Stefano. Port and node support to compact routing. Technical Report R.97-17, Dipartimento di Ingegneria Elettrica, University di L'Aquila (L'Aquila, Italy), 1997. Submitted for pubblication.Google Scholar
  8. 8.
    W. H. Cunningham. Decomposition of directed graphs. SIAM Journal on Alg. Disc. Meth., 3:214–228, 1982.Google Scholar
  9. 9.
    E. Dahlhaus. An efficient parallel recognition algorithm for parity graphs. In O. Abou-Rabia et al., editor, ICCI '93, pages 82–86, 1993.Google Scholar
  10. 10.
    E. Dahlhaus. Efficient parallel and linear time sequential split decomposition.In FSTTCS: Foundations of Software Technology and Theoretical Computer Science. Lecture Notes in Computer Science, vol. 880, 1994.Google Scholar
  11. 11.
    G. Di Stefano. A routing algorithm for networks based on distance-hereditary topologies. In SIROCCO '96: The 3rd Int. Colloquium on Structural Information and Communication Complexity, 1996.Google Scholar
  12. 12.
    P. L. Hammer and F. Maffray. Completely separable graphs. DAMATH: Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science, 27, 1990.Google Scholar
  13. 13.
    F. Harary. Graph Theory. Addison-Wesley, 1969.Google Scholar
  14. 14.
    L. Hellerstein and M. Kaxpinski.Learning read-once formulas using membership queries. In Proc. of the Second Annual YVorkshop on Computational Learning Theory, pages 146–161. Morgan Kaufmann, 1989.Google Scholar
  15. 15.
    J. E. Hopcroft and R. M. Karp. An O(n 5/2) algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing, 2(4):225–231, December 1973.Google Scholar
  16. 16.
    T. Przytycka and D. G. Corneil. Parallel algorithms for parity graphs. Journal of Algorithms, 12(1):96–109, March 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Serafino Cicerone
    • 1
  • Gabriele Di Stefano
    • 1
  1. 1.Dipartimento di Ingegneria ElettricaUniversità dell'AquilaL'AquilaItaly

Personalised recommendations