Planar Graphs: Logical Complexity and Parallel Isomorphism Tests

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.

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