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Abstract

We investigate the s-t-connectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspace-reducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the planar case.

We also consider a previously-studied subclass of planar graphs known as grid graphs. We show that the directed planar s-t-connectivity problem reduces to the reachability problem for directed grid graphs.

A special case of the grid-graph reachability problem where no edges are directed from right to left is known as the “layered grid graph reachability problem”. We show that this problem lies in the complexity class UL.

Keywords

Planar Graph Coarse Grid Outer Face Grid Graph Reachability 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.

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References

  1. [AM04]
    Allender, E., Mahajan, M.: The complexity of planarity testing. Information and Computation 189, 117–134 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  2. [ARZ99]
    Allender, E., Reinhardt, K., Zhou, S.: Isolation, matching, and counting: Uniform and nonuniform upper bounds. Journal of Computer and System Sciences 59(2), 164–181 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [Bar02]
    Barrington, D.A.M.: Grid graph reachability problems. Talk presented at Dagstuhl Seminar on Complexity of Boolean Functions, Seminar number 02121 (2002)Google Scholar
  4. [BH91]
    Buss, S.R., Hay, L.: On truth-table reducibility to SAT. Inf. Comput. 91(1), 86–102 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [BK78]
    Blum, M., Kozen, D.: On the power of the compass (or, why mazes are easier to search than graphs). In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 132–142 (1978)Google Scholar
  6. [BLMS98]
    Barrington, D.A.M., Lu, C.-J., Miltersen, P.B., Skyum, S.: Searching constant width mazes captures the AC0 hierarchy. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 73–83. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  7. [BST93]
    Buhrman, H., Spaan, E., Torenvliet, L.: The relative power of logspace and polynomial time reductions. Computational Complexity 3, 231–244 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [Ete97]
    Etessami, K.: Counting quantifiers, successor relations, and logarithmic space. Journal of Computer and System Sciences 54(3), 400–411 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [GT87]
    Gross, J., Tucker, T.: Topological Graph Theory, 1st edn. John Wiley and Sons, Chichester (1987)zbMATHGoogle Scholar
  10. [Imm88]
    Immerman, N.: Nondeterministic space is closed under complementation. SIAM Journal on Computing 17, 935–938 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  11. [JLR01]
    Jakoby, A., Liskiewicz, M., Reischuk, R.: Space efficient algorithms for seriesparalle graphs. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 339–352. Springer, Heidelberg (2001) (to appear in J. Algorithms)CrossRefGoogle Scholar
  12. [Lan97]
    Lange, K.-J.: An unambiguous class possessing a complete set. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200, pp. 339–350. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  13. [LL76]
    Ladner, R., Lynch, N.: Relativization of questions about log space reducibility. Mathematical Systems Theory 10, 19–32 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  14. [MT01]
    Mohar, B., Thomassen, C.: Graphs on surfaces, 1st edn. John Hopkins University Press, Maryland (2001)zbMATHGoogle Scholar
  15. [MV00]
    Mahajan, M., Varadarajan, K.R.: A new NC-algorithm for finding a perfect matching in bipartite planar and small genus graphs. In: ACM Symposium on Theory of Computing (STOC), pp. 351–357 (2000)Google Scholar
  16. [NTS95]
    Nisan, N., Ta-Shma, A.: Symmetric Logspace is closed under complement. Chicago Journal of Theoretical Computer Science (1995)Google Scholar
  17. [RA00]
    Reinhardt, K., Allender, E.: Making nondeterminism unambiguous. SIAM Journal of Computing 29, 1118–1131 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  18. [Rei05]
    Reingold, O.: Undirected st-connectivity in log-space. In: Proceedings 37th Symposium on Foundations of Computer Science, pp. 376–385. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
  19. [Sze88]
    Szelepcsényi, R.: The method of forced enumeration for nondeterministic automata. Acta Informatica 26, 279–284 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  20. [Tho90]
    Thomassen, C.: Embeddings of graphs with no short noncontractible cycles. J. Comb. Theory Ser. B 48(2), 155–177 (1990)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Eric Allender
    • 1
  • Samir Datta
    • 2
  • Sambuddha Roy
    • 1
  1. 1.Department of Computer ScienceRutgers UniversityPiscatawayUSA
  2. 2.Chennai Mathematical InstituteChennaiIndia

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