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.
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Verbitsky, O.: Planar graphs: Logical complexity and parallel isomorphism tests. E-print (2006), http://arxiv.org/abs/cs.CC/0607033
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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
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DOI: https://doi.org/10.1007/978-3-540-70918-3_58
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