Abstract
We prove that every triconnected planar graph on n vertices is definable by a first order sentence that uses at most 15 variables and has quantifier depth at most 11 log2 n + 45. As a consequence, a canonic form of such graphs is computable in AC1 by the 14-dimensional Weisfeiler-Lehman algorithm. This gives us another AC1 algorithm for the planar graph isomorphism.
Supported by an Alexander von Humboldt fellowship.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Cai, J.-Y., Fürer, M., Immerman, N.: An optimal lower bound on the number of variables for graph identification. Combinatorica 12, 389–410 (1992)
Diestel, R.: Graph theory. Springer, Heidelberg (2006)
Grohe, M.: Fixed-point logics on planar graphs. In: Proc. of the Ann. Conf. on Logic in Computer Science, pp. 6–15 (1998)
Grohe, M., Marino, J.: Definability and descriptive complexity on databases of bounded tree-width. In: Beeri, C., Bruneman, P. (eds.) ICDT 1999. LNCS, vol. 1540, pp. 70–82. Springer, Heidelberg (1998)
Grohe, M., Verbitsky, O.: Testing graph isomorphism in parallel by playing a game. In: Bugliesi, M., et al. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 3–14. Springer, Heidelberg (2006)
Hopcroft, J.E.: An n logn algorithm for isomorphism of planar triply connected graphs. Technical Report CS-192. Stanford University (1971)
Hopcroft, J.E., Tarjan, R.E.: Isomorphism of planar graphs (working paper). In: Complexity of computer computations, pp. 131–152. Plenum Press, New York (1972)
Hopcroft, J.E., Tarjan, R.E.: Deviding a graph into triconnected components. SIAM Journal on Computing 2, 135–158 (1973)
Immerman, N.: Descriptive complexity. Springer, Heidelberg (1999)
Kim, J.-H., et al.: How complex are random graphs in first order logic? Random Structures and Algorithms 26, 119–145 (2005)
Miller, G.L., Reif, J.H.: Parallel tree contraction. Part 2: further applications. SIAM J. Comput. 20, 1128–1147 (1991)
Pikhurko, O., Spencer, J., Verbitsky, O.: Succinct definitions in first order graph theory. Annals of Pure and Applied Logic 139, 74–109 (2006)
Pikhurko, O., Veith, H., Verbitsky, O.: First order definability of graphs: tight bounds on quantifier rank. Discrete Applied Mathematics 154, 2511–2529 (2006)
Ramachandran, V., Reif, J.: Planarity testing in parallel. J. Comput. Syst. Sci. 49, 517–561 (1994)
Weisfeiler, B., Yu., L.A.A.: A reduction of a graph to a canonical form and an algebra arising during this reduction (in Russian). Nauchno-Technicheskaya Informatsia, Seriya 2. 9, 12–16 (1968)
Verbitsky, O.: Planar graphs: Logical complexity and parallel isomorphism tests. E-print (2006), http://arxiv.org/abs/cs.CC/0607033
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Verbitsky, O. (2007). Planar Graphs: Logical Complexity and Parallel Isomorphism Tests. In: Thomas, W., Weil, P. (eds) STACS 2007. STACS 2007. Lecture Notes in Computer Science, vol 4393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70918-3_58
Download citation
DOI: https://doi.org/10.1007/978-3-540-70918-3_58
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70917-6
Online ISBN: 978-3-540-70918-3
eBook Packages: Computer ScienceComputer Science (R0)