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Lin Chen, "Graph Isomorphism and Identification Matrices: Parallel Algorithms," IEEE Transactions on Parallel and Distributed Systems, vol. 7, no. 3, pp. 308319, March, 1996.  
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@article{ 10.1109/71.491584, author = {Lin Chen}, title = {Graph Isomorphism and Identification Matrices: Parallel Algorithms}, journal ={IEEE Transactions on Parallel and Distributed Systems}, volume = {7}, number = {3}, issn = {10459219}, year = {1996}, pages = {308319}, doi = {http://doi.ieeecomputersociety.org/10.1109/71.491584}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Parallel and Distributed Systems TI  Graph Isomorphism and Identification Matrices: Parallel Algorithms IS  3 SN  10459219 SP308 EP319 EPD  308319 A1  Lin Chen, PY  1996 KW  Consecutive and circular 1s properties KW  graph isomorphism testing KW  identification matrices KW  parallel algorithms KW  PRAM KW  time and processor requirements. VL  7 JA  IEEE Transactions on Parallel and Distributed Systems ER   
Abstract—In this paper, we explore some properties of identification matrices and exhibit some uses of identification matrices in studying the graph isomorphism problem, a famous open problem. We show that, given two graphs in the form of a certain identification matrix, isomorphism can be tested efficiently in parallel if at least one matrix satisfies the circular 1s property, and more efficiently in parallel if at least one matrix satisfies the consecutive 1s property. Graphs which have identification matrices satisfying the consecutive 1s property include, among others, proper interval graphs and doubly convex bipartite graphs. The result presented here substantially broadens the class of graphs for which there are known efficient parallel isomorphism testing algorithms.
[1] G.S. Adhar and S. Peng, "Parallel Algorithms for Cographs and Parity Graphs with Applications," J. Algorithms, vol. 11, no. 2, pp. 252284, June 1990.
[2] K.S. Booth and C.J. Colbourn, "Problems Polynomially Equivalent to Graph Isomorphism," Technical Report CS7704, Computer Science Dept., Univ. of Waterloo, Canada, 1979.
[3] K.S. Booth and G.S. Lueker, "Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQTree Algorithms," J. Computer and System Sciences, vol. 13, pp. 335379, 1976.
[4] L. Chen, "Efficient Parallel Algorithms for Several Intersection Graphs," Proc. 22nd Int'l Symp. Circuits and Systems, pp. 973976. IEEE, 1989.
[5] T. Yabe et al., "A Configurable DRAM Macro Design for 2112 Derivative Organizations to Be Synthesized Using a Memory Generator," Digest of Technical Papers, IEEE Int'l SolidState Circuits Conf., IEEE, Piscataway, N.J., 1998, pp. 7273.
[6] L. Chen, "Testing Isomorphism for Transformable Convex Bipartite Graphs in Polynomial Time," Technical Report OSUCISRC12/89TR54, Dept. of Computer and Information Science, College of Eng., Ohio State Univ., 1989.
[7] L. Chen, "Parallel Complexity of Discrete Problems," PhD thesis, Ohio State Univ., 1990.
[8] L. Chen, "Efficient Parallel Recognition of Some Circular Arc Graphs, I," Algorithmica, vol. 9, no. 3, pp. 217238, Mar. 1993.
[9] L. Chen, "Parallel Graph Isomorphism Detection with Identification Matrices," Proc. Int'l Symp. Parallel Architectures, Algorithms, and Networks. IEEE, 1994.
[10] L. Chen, "Revisiting Circular Arc Graphs," Proc. Fifth Ann. Int'l Symp. Algorithms and Computation, D.Z. Du and X.S. Zhang, eds., Lecture Notes in Computer Science, vol. 834, pp. 559566. SpringerVerlag, 1994.
[11] L. Chen and Y. Yesha, "Parallel Recognition of the Consecutive Ones Property with Applications," J. Algorithms, vol. 12, no. 3, pp. 375392, Sept. 1991.
[12] L. Chen and Y. Yesha, "Efficient Parallel Algorithms for Bipartite Permutation Graphs," Networks, vol. 23, no. 1, pp. 2939, Jan. 1993.
[13] E. Dekel and S. Sahni, "A Parallel Matching Algorithm for Convex Bipartite Graphs and Applications to Scheduling," J. Parallel and Distributed Computing, vol. 1, pp. 185205, 1984.
[14] D.R. Fulkerson and O.A. Gross, "Incidence Matrices and Interval Graphs," Pacific J. Mathematics, vol. 15, pp. 835855, 1965.
[15] F. Gavril, "Algorithms on CircularArc Graphs," Networks, vol. 4, pp. 357369, 1974.
[16] P.C. Gilmore and A.J. Hoffman, "A Characterization of Comparability Graphs and of Interval Graphs," Canadian J. Mathematics, vol. 16, pp. 539548, 1964.
[17] X. He and Y. Yesha, "Parallel Recognition and Decomposition of Two Terminal Series Parallel Graphs," Information and Computation, vol. 75, pp. 1538, 1987.
[18] C.W. Ho and R.C.T. Lee, "Efficient Parallel Aalgorithms for Finding Maximal Cliques, Clique Trees, and Minimum Coloring on Chordal Graphs," Information Processing Letters, vol. 28, pp. 301309, Aug. 1988.
[19] C.M. Hoffmann, GroupTheoretic Algorithms and Graph Isomorphism, vol. 136, Lecture Notes in Computer Science. SpringerVerlag, 1982.
[20] J.E. Hopcroft and J.K. Wong, "Linear Time Algorithm for Isomorphism of Planar Graphs," Proc. Sixth ACM Ann. Symp. Theory of Computing, pp. 172184, 1974.
[21] R.M. Karp and V. Ramachandran, "Parallel Algorithms for SharedMemory Machines," Handbook of Theoretical Computer Science, J. van Leeuwen, ed., vol. A, pp. 869941.Amsterdam: NorthHolland, 1990.
[22] W. Lipski Jr. and F.P. Preparata, "Efficient Algorithms for Finding Maximum Matchings in Convex Bipartite Graphs and Related Problems," Acta Informatica, vol. 15, no. 4, pp. 329346, Aug. 1981.
[23] G.L. Miller and J.H. Reif, "Parallel Tree Contraction, Part 1: Fundamentals," Randomness and Computation, S. Micali, ed., vol. 5, pp. 4772. JAI Press, 1989.
[24] G.L. Miller and J.H. Reif, "Parallel Tree Contraction, Part 2: Further Applications," SIAM J. Computing, vol. 20, no. 6, pp. 1,1281,147, Dec. 1991.
[25] F.S. Roberts, "Representations of Indifference Relations," PhD thesis, Stanford Univ., 1968.
[26] J. Spinrad, A. Brandstädt, and L. Stewart, "Bipartite Permutation Graphs," Discrete Applied Mathematics, vol. 18, pp. 279292, 1987.
[27] R.E. Tarjan and U. Vishkin, "An Efficient Parallel Biconnectivity Algorithm," SIAM J. Computing, vol. 14, no. 4, pp. 862874, Nov. 1985.
[28] A.C. Tucker, "Matrix Characterization of CircularArc Graphs," Pacific J. Mathematics, vol. 39, no. 2, pp. 535545, 1971.
[29] T. Wu, "An O(n3) Isomorphism Test for CircularArc Graphs," PhD thesis, State Univ. of New York, Stony Brook, 1983.