Planar Graphs: Logical Complexity and Parallel Isomorphism Tests

Abstract

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 $11{\log }_{2}n+43$. As a consequence, a canonic form of such graphs is computable in $A{C}^1$ by the 14-dimensional Weisfeiler-Lehman algorithm. This provides another way to show that the planar graph isomorphism is solvable in $A{C}^1$.

36 pages