A selected tour of the theory of identification matrices

  • Lin Chen
Session 7: Graph Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1276)


A selected tour of the theory of identification matrices is offered here. We show that shortest-path adjacency matrices are identification matrices for all simple graphs and adjacency matrices are identification matrices for all bipartite graphs. Additionally, we provide an improved proof that augmented adjacency matrices satisfying the circular 1's property are identification matrices. In passing, we present a matrix characterization of doubly convex bipartite graphs. As an application of the theory of identification matrices, we describe an improved method for testing isomorphism of Γ circular arc graphs.


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  1. 1.
    K. S. Booth and G. S. Lueker. Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. Journal of Computer and System Sciences, 13:335–379, 1976.Google Scholar
  2. 2.
    L. Chen. Parallel graph isomorphism detection with identification matrices. In S. Horiguchi, D. F. Hsu, and M. Kimura, editors, Proc. Int'l Symp. on Parallel Architectures, Algorithms and Networks, pages 105–112. IEEE, 1994.Google Scholar
  3. 3.
    L. Chen. Sequential graph isomorphism detection with identification matrices. In Proceedings, 4th International Conference for Young Computer Scientists, 1995.Google Scholar
  4. 4.
    L. Chen. Graph isomorphism and identification matrices: Parallel algorithms. IEEE Transactions on Parallel and Distributed Systems, 7(3):308–319, March 1996.Google Scholar
  5. 5.
    L. Chen. Efficient parallel recognition of some circular arc graphs, II. Algorithmica, 17(3):266–280, March 1997.Google Scholar
  6. 6.
    L. Chen and Y. Yesha. Parallel recognition of the consecutive ones property with applications. J. Algorithms, 12(3):375–392. September 1991.Google Scholar
  7. 7.
    L. Chen and Y. Yesha. Efficient parallel algorithms for bipartite permutation graphs. Networks, 23(1):29–39, January 1993.Google Scholar
  8. 8.
    M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., San Francisco, 1979.Google Scholar
  9. 9.
    W.-L. Hsu. O(mn) algorithms for the recognition and isomorphism problems on circular-arc graphs. SIAM Journal on Computing, 24(3):411–439, June 1995.Google Scholar
  10. 10.
    R. M. Karp. Reducibility among combinatorial problems. In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations. Plenum Press, New York, 1972.Google Scholar
  11. 11.
    R. M. Karp and V. Ramachandran. Parallel algorithms for shared-memory machines. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, Vol. A, pages 869–941. North Holland, Amsterdam, 1990.Google Scholar
  12. 12.
    G. S. Lueker and K. S. Booth. A linear time algorithm for deciding interval graph isomorphism. Journal of the ACM, 26(2):183–195, April 1979.Google Scholar
  13. 13.
    F. S. Roberts. Representations of Indifference Relations. PhD thesis, Stanford University, 1968.Google Scholar
  14. 14.
    T. Takaoka. A new upper bound on the complexity of the all pairs shortest path problem. Information Processing Letters, 43(4):195–199, September 1992.Google Scholar
  15. 15.
    A. C. Tucker. Matrix characterization of circular-arc graphs. Pacific J. Math., 39(2):535–545, 1971.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Lin Chen
    • 1
  1. 1.FRLLos Angeles

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