Planar Graphs: Logical Complexity and Parallel Isomorphism Tests

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We prove that every triconnected planar graph is definable by a first order sentence that uses at most 15 variables and has quantifier depth at most 11₂ n+43. As a consequence, a canonic form of such graphs is computable in AC¹ by the 14-dimensional Weisfeiler-Lehman algorithm. This provides another way to show that the planar graph isomorphism is solvable in AC¹.