Martin Grohe
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We give a logical characterization of the polynomial-time properties of graphs embeddable in some surface. For every surface S, a property P of graphs embeddable in S is decidable in polynomial time if and only if it is definable in fixed-point logic with counting. It is a consequence of this result that for every surface S there is a k such that a simple combinatorial algorithm, namely “the k-dimensional Weisfeiler-Lehman algorithm”, decides isomorphism of graphs embeddable in S in polynomial time.
We also present (without proof) generalizations of these results to arbitrary classes of graphs with excluded minors.
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- Babai, L. 1981. Moderately exponential bound for graph isomorphism. In Fundamentals of Computation Theory (FCT'81), F. Gécseg, Ed., Lecture Notes in Computer Science, vol. 117, Springer, 34--50. Google Scholar
Digital Library
- Babai, L., Erdös, P., and Selkow, S. 1980. Random graph isomorphism. SIAM J. Comput. 9, 628--635.Google ScholarDigital Library
- Babai, L. and Luks, E. 1983. Canonical labeling of graphs. In Proceedings of the 15th ACM Symposium on Theory of Computing. 171--183. Google ScholarDigital Library
- Blass, A., Gurevich, Y., and Shelah, S. 1999. Choiceless polynomial time. Ann. Pure Appl. Logic 100, 141--187.Google ScholarCross Ref
- Blass, A., Gurevich, Y., and Shelah, S. 2002. On polynomial time computation over unordered structures. J. Symbol. Logic 67, 1093--1125.Google ScholarCross Ref
- Bodlaender, H. 1990. Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees. J. Algor. 11, 631--643. Google ScholarDigital Library
- Cai, J., Fürer, M., and Immerman, N. 1992. An optimal lower bound on the number of variables for graph identification. Combinatorica 12, 389--410.Google ScholarCross Ref
- Chandra, A. and Harel, D. 1982. Structure and complexity of relational queries. J. Comput. Syst. Sci. 25, 99--128.Google ScholarCross Ref
- Conway, J. and Gordon, C. 1983. Knots and links in spatial graphs. J. Graph Theory 7, 445--453.Google ScholarCross Ref
- Courcelle, B. and Olariu, S. 2000. Upper bounds to the clique-width of graphs. Discr. Appl. Math. 101, 77--114. Google ScholarDigital Library
- Dawar, A., Grohe, M., Holm, B., and Laubner, B. 2009. Logics with rank operators. In Proceedings of the 24th IEEE Symposium on Logic in Computer Science. 113--122. Google ScholarDigital Library
- Demaine, E., Hajiaghayi, M., and Kawarabayashi, K. 2005. Algorithmic graph minor theory: Decomposition, approximation, and coloring. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science. 637--646. Google ScholarDigital Library
- Diestel, R. 2005. Graph Theory 3rd Ed. Springer.Google Scholar
- Ebbinghaus, H.-D. and Flum, J. 1999. Finite Model Theory, 2nd Ed. Springer.Google Scholar
- Ebbinghaus, H.-D., Flum, J., and Thomas, W. 1994. Mathematical Logic 2nd Ed. Springer.Google Scholar
- Fagin, R. 1974. Generalized first--order spectra and polynomial--time recognizable sets. In SIAM-AMS Proceedings on Complexity of Computation. Vol. 7, R. Karp, Ed., 43--73.Google Scholar
- Filotti, I. S. and Mayer, J. N. 1980. A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus. In Proceedings of the 12th ACM Symposium on Theory of Computing. 236--243. Google ScholarDigital Library
- Flum, J. and Grohe, M. 2000. On fixed-point logic with counting. J. Symbol. Logic 65, 777--787.Google ScholarCross Ref
- Grädel, E., Kolaitis, P., Libkin, L., Marx, M., Spencer, J., Vardi, M., Venema, Y., and Weinstein, S. 2007. Finite Model Theory and Its Applications. Springer.Google Scholar
- Grädel, E. and Otto, M. 1993. Inductive definability with counting on finite structures. In Proceedings of the Computer Science Logic, 6th Workshop (CSL'92). Selected Papers, E. Börger, G. Jäger, H. K. Büning, S. Martini, and M. Richter, Eds., Lecture Notes in Computer Science, vol. 702, Springer, 231--247. Google ScholarDigital Library
- Grohe, M. 1998. Fixed-Point logics on planar graphs. In Proceedings of the 13th IEEE Symposium on Logic in Computer Science. 6--15. Google ScholarDigital Library
- Grohe, M. 2000. Isomorphism testing for embeddable graphs through definability. In Proceedings of the 32nd ACM Symposium on Theory of Computing. 63--72. Google ScholarDigital Library
- Grohe, M. 2008. Definable tree decompositions. In Proceedings of the 23rd IEEE Symposium on Logic in Computer Science. 406--417. Google ScholarDigital Library
- Grohe, M. 2010. Fixed-Point definability and polynomial time on graphs with excluded minors. In Proceedings of the 25th IEEE Symposium on Logic in Computer Science. Google ScholarDigital Library
- Grohe, M. 2011. From polynomial time queries to graph structure theory. Comm. ACM 54, 6, 104--112. Google ScholarDigital Library
- Grohe, M. 2012. Descriptive complexity, canonisation, and definable graph structure theory. http://www2. informatik.hu-berlin.de/~grohe/cap.Google Scholar
- Grohe, M. and Mariño, J. 1999. Definability and descriptive complexity on databases of bounded tree-width. In Proceedings of the 7th International Conference on Database Theory. C. Beeri and P. Buneman, Eds., Lecture Notes in Computer Science, vol. 1540, Springer, 70--82. Google ScholarDigital Library
- Grohe, M. and Verbitsky, O. 2006. Testing graph isomorphism in parallel by playing a game. In Proceedings of the 33rd International Colloquium on Automata, Languages and Programming, Part I, M. Bugliesi, B. Preneel, V. Sassone, and I. Wegener, Eds., Lecture Notes in Computer Science, vol. 4051, Springer, 3--14. Google ScholarDigital Library
- Gross, J. and Tucker, T. 1987. Topological Graph Theory. Wiley. Google ScholarDigital Library
- Gurevich, Y. 1984. Toward logic tailored for computational complexity. In Computation and Proof Theory, M. M. Richter, E. Börger, W. Oberschelp, B. Schinzel, and W. Thomas, Eds., Lecture Notes in Mathematics, vol. 1104, Springer, 175--216.Google ScholarCross Ref
- Gurevich, Y. 1988. Logic and the challenge of computer science. In Current Trends in Theoretical Computer Science, E. Börger, Ed., Computer Science Press, 1--57.Google Scholar
- Gurevich, Y. and Shelah, S. 1986. Fixed point extensions of first--order logic. Ann. Pure Appl. Logic 32, 265--280.Google ScholarCross Ref
- Hella, L., Kolaitis, P., and Luosto, K. 1996. Almost everywhere equivalence of logics in finite model theory. Bull. Symbol. Logic 2, 422--443.Google ScholarCross Ref
- Hopcroft, J. and Tarjan, R. 1972. Isomorphism of planar graphs (working paper). In Complexity of Computer Computations, R. E. Miller and J. W. Thatcher, Eds., Plenum Press.Google Scholar
- Hopcroft, J. and Wong, J. 1974. Linear time algorithm for isomorphism of planar graphs. In Proceedings of the 6th ACM Symposium on Theory of Computing. 172--184. Google ScholarDigital Library
- Immerman, N. 1982. Relational queries computable in polynomial time (extended abstract). In Proceedings of the 14th ACM Symposium on Theory of Computing. 147--152. Google ScholarDigital Library
- Immerman, N. 1986. Relational queries computable in polynomial time. Inf. Control 68, 86--104. Google ScholarDigital Library
- Immerman, N. 1987. Expressibility as a complexity measure: Results and directions. In Proceedings of the 2nd IEEE Symposium on Structure in Complexity Theory. 194--202.Google Scholar
- Immerman, N. 1999. Descriptive Complexity. Springer.Google Scholar
- Immerman, N. and Lander, E. 1990. Describing graphs: A first-order approach to graph canonization. In Complexity Theory Retrospective, A. Selman, Ed., Springer, 59--81.Google Scholar
- Köbler, J. and Verbitsky, O. 2008. From invariants to canonization in parallel. In Proceedings of the 3rd International Computer Science Symposium. E. Hirsch, A. Razborov, A. Semenov, and A. Slissenko, Eds., Lecture Notes in Computer Science, vol. 5010, Springer, 216--227. Google ScholarDigital Library
- Kreutzer, S. 2002. Expressive equivalence of least and inflationary fixed-point logic. In Proceedings of the 17th IEEE Symposium on Logic in Computer Science. 403--413. Google ScholarDigital Library
- Laubner, B. 2010. Capturing polynomial time on interval graphs. In Proceedings of the 25th IEEE Symposium on Logic in Computer Science. 199--208. Google ScholarDigital Library
- Levine, H. 1963. Homotopic curves on surfaces. Proc. Amer. Math. Soc. 14, 986--990.Google ScholarCross Ref
- Libkin, L. 2004. Elements of Finite Model Theory. Springer. Google ScholarDigital Library
- Luks, E. 1982. Isomorphism of graphs of bounded valance can be tested in polynomial time. J. Comput. Syst. Sci. 25, 42--65.Google ScholarCross Ref
- Miller, G. 1983. Isomorphism of graphs which are pairwise k-separable. Inf. Control 56, 21--33.Google ScholarCross Ref
- Miller, G. L. 1980. Isomorphism testing for graphs of bounded genus. In Proceedings of the 12th ACM Symposium on Theory of Computing. 225--235. Google ScholarDigital Library
- Mohar, B. 1999. A linear time algorithm for embedding graphs in an arbitrary surface. SIAM J. Discr. Math. 12, 6--26. Google ScholarDigital Library
- Mohar, B. and Thomassen, C. 2001. Graphs on Surfaces. Johns Hopkins University Press.Google Scholar
- Otto, M. 1997. Bounded Variable Logics and Counting—A Study in Finite Models. Lecture Notes in Logic, vol. 9, Springer.Google Scholar
- Oum, S.-I. and Seymour, P. 2006. Approximating clique-width and branch-width. J. Combin. Theory, Series B 96, 514--528. Google ScholarDigital Library
- Ponomarenko, I. N. 1988. The isomorphism problem for classes of graphs that are invariant with respect to contraction. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 174, Teor. Slozhn. Vychisl. 3, 147--177, 182. In Russian.Google Scholar
- Ringel, G. 1974. Map Color Theorem. Springer.Google Scholar
- Robertson, N. and Seymour, P. 1991. Graph minors X. Obstructions to tree-decomposition. J. Combin. Theory, Series B 52, 153--190. Google ScholarDigital Library
- Robertson, N. and Seymour, P. 1995. Graph minors XIII. The disjoint paths problem. J. Combin. Theory, Series B 63, 65--110. Google ScholarDigital Library
- Robertson, N. and Seymour, P. 1999. Graph minors XVI. Excluding a non-planar graph. J. Combin. Theory, Series B 77, 1--27. Google ScholarDigital Library
- Robertson, N. and Seymour, P. 2004. Graph minors XX. Wagner's conjecture. J. Combin. Theory, Series B 92, 325--357. Google ScholarDigital Library
- Robertson, N. and Vitray, R. 1990. Representativity of surface embeddings. In Paths, Flows and VLSI-Layout, B. Korte, L. Lovász, H. Prömel, and A. Schrijver, Eds., Springer, 293--328.Google Scholar
- Thomassen, C. 1988. The graph genus problem is NP-complete. J. Algor. 10, 458--576. Google ScholarDigital Library
- Vardi, M. 1982. The complexity of relational query languages. In Proceedings of the 14th ACM Symposium on Theory of Computing. 137--146. Google ScholarDigital Library
- Verbitsky, O. 2007. Planar graphs: Logical complexity and parallel isomorphism tests. In Proceedings of the 24th Annual Symposium on Theoretical Aspects of Computer Science. W. Thomas and P. Weil, Eds., Lecture Notes in Computer Science, vol. 4393, Springer, 682--693. Google ScholarDigital Library
- Whitney, H. 1932. Congruent graphs and the connectivity of graphs. Amer. J. Math. 54, 150--168.Google ScholarCross Ref
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