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Algorithmic graph embeddings

Extended abstract
  • Jianer Chen
Session 3A: Graph Algorithms
  • 137 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 959)

Abstract

The complexity of embedding a graph into a variety of topological surfaces is investigated. A new data structure for graph embeddings is introduced and shown to be superior to the previously known data structures. In particular, the new data structure efficiently supports all on-line operations for general graph embeddings. Based on this new data structure, very efficient algorithms are developed to solve the problem “given a graph G and an integer k, construct a genus k embedding for the graph G” for a large range of the integers k and for a large class of graphs.

Keywords

Planar Graph Average Genus Rotation System Linear Time Algorithm Interior Vertex 
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.

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References

  1. 1.
    Aho, A. V., Hopcroft, J. E., and Ullman, J. D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, (1974)Google Scholar
  2. 2.
    Chen, J.: A linear time algorithm for isomorphism of graphs of bounded average genus. SIAM J. on Discrete Mathematics 7 (1994) 614–631CrossRefGoogle Scholar
  3. 3.
    Chen, J., Archdeacon, D., and Gross, J. L.: Maximum genus and connectivity. Discrete Mathematics (1995) to appearGoogle Scholar
  4. 4.
    Chen, J., Gross, J. L.: Limit points for average genus I. 3-connected and 2-connected simplicial graphs. J. Comb. Theory Ser. B 55 (1992) 83–103CrossRefGoogle Scholar
  5. 5.
    Chen, J., Gross, J. L.: Limit points for average genus II. 2-connected non-simplicial graphs. J. Comb. Theory Ser. B 56 (1992) 108–129CrossRefGoogle Scholar
  6. 6.
    Chen, J., Gross, J. L.: Kuratowski-type theorems for average genus. J. Comb. Theory Ser. B 57 (1993) 100–121CrossRefGoogle Scholar
  7. 7.
    Chen, J. and Kanchi, S. P.: Graph embeddings and graph ear decompositions. Lecture Notes in Computer Science 790 (1994) 376–387Google Scholar
  8. 8.
    Chen, J., Kanchi, S. P., and Kanevsky, A.: On the complexity of graph embeddings. Lecture Notes in Computer Science 709 (1993) 234–245Google Scholar
  9. 9.
    Di Battista, G., Eades, P., and Tamassia, R.: Algorithms for automatic graph drawing: an annotated bibliography. Tech. Report (1993) Dept. Computer Science, Brown UniversityGoogle Scholar
  10. 10.
    Djidjev, H. and Reif, J.: An efficient algorithm for the genus problem with explicit construction of forbidden subgraphs. Proc. 23rd Annual ACM Symposium on Theory of Computing (1991) 337–347Google Scholar
  11. 11.
    Filotti, I. S.: An algorithm for imbedding cubic graphs in the torus. Journal of Computer and System Sciences 20 (1980) 255–276CrossRefGoogle Scholar
  12. 12.
    Filotti, I. S., Miller, G. L., and Reif, J. H.: On determining the genus of a graph in O(vO(g)) steps. Proc. 11th Annual ACM Symp. on Theory of Comput. (1979) 27–37Google Scholar
  13. 13.
    Frederickson, G. N.: Using cellular graph embeddings in solving all pairs shortest paths problems. Proc. 30th IEEE Symp. on Foundations of Computer Science (1989) 448–453Google Scholar
  14. 14.
    Frederickson, G. N. and Janardan, R.: Designing networks with compact routing tables. Algorithmica 3 (1988) 171–190CrossRefMathSciNetGoogle Scholar
  15. 15.
    Furst, M. L., Gross, J. L., and McGeoch, L. A.: Finding a maximum-genus graph imbedding. Journal of ACM 35-3 (1988) 523–534CrossRefGoogle Scholar
  16. 16.
    Garey, M. R. and Johnson, D. S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979Google Scholar
  17. 17.
    Gross, J. L. and Furst, M. L.: Hierarchy for imbedding-distribution invariants of a graph. J. Graph Theory 11 (1987) 205–220Google Scholar
  18. 18.
    Gross, J. L., Tucker, T. W.: Topological Graph Theory. Wiley-Interscience, New York (1987)Google Scholar
  19. 19.
    Hopcroft, J. E. and Tarjan, R. E.: Efficient planarity testing. Journal of the ACM 21 (1974) 549–568.CrossRefGoogle Scholar
  20. 20.
    Italiano, G. F., La Poutre, J. A., and Rauch, M. H.: Fully dynamic planarity testing in planar embedded graphs. Lecture Notes in Computer Science 726 (1993) 576–590.Google Scholar
  21. 21.
    Mohar, B.: Projective planarity in linear time. Journal of Algorithms 15 (1993) 482–502CrossRefGoogle Scholar
  22. 22.
    Preparata, F. P. and Shamos, M. I.: Computational Geometry: An Introduction. Springer-Verlag 1985Google Scholar
  23. 23.
    Tamassia, R.: A dynamic data structure for planar graph embedding. Lecture Notes in Computer Science 317 (1988) 576–590.Google Scholar
  24. 24.
    Thomassen, C.: The graph genus problem is NP-complete. Journal of Algorithms 10 (1989) 568–576CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Jianer Chen
    • 1
  1. 1.Department of Computer ScienceTexas A&M UniversityCollege StationUSA

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