  The Weisfeiler-Leman (WL) algorithm is a well-known combinatorial procedure
for detecting symmetries in graphs and it is widely used in graph-isomorphism
tests. It proceeds by iteratively refining a colouring of vertex tuples. The
number of iterations needed to obtain the final output is crucial for the
parallelisability of the algorithm.
  We show that there is a constant k such that every planar graph can be
identified (that is, distinguished from every non-isomorphic graph) by the
k-dimensional WL algorithm within a logarithmic number of iterations. This
generalises a result due to Verbitsky (STACS 2007), who proved the same for
3-connected planar graphs.
  The number of iterations needed by the k-dimensional WL algorithm to identify
a graph corresponds to the quantifier depth of a sentence that defines the
graph in the (k+1)-variable fragment C^{k+1} of first-order logic with counting
quantifiers. Thus, our result implies that every planar graph is definable with
a C^{k+1}-sentence of logarithmic quantifier depth.
