  The Weisfeiler-Leman (WL) algorithm is a combinatorial procedure that
computes colorings on graphs, which can often be used to detect their
(non-)isomorphism. Particularly the 1- and 2-dimensional versions 1-WL and 2-WL
have received much attention, due to their numerous links to other areas of
computer science.
  Knowing the expressive power of a certain dimension of the algorithm usually
amounts to understanding the computed colorings. An increase in the dimension
leads to finer computed colorings and, thus, more graphs can be distinguished.
For example, on the class of planar graphs, 3-WL solves the isomorphism
problem. However, the expressive power of 2-WL on the class is poorly
understood (and, in particular, it may even well be that it decides
isomorphism).
  In this paper, we investigate the colorings computed by 2-WL on planar
graphs. Towards this end, we analyze the graphs induced by edge color classes
in the graph. Based on the obtained classification, we show that for every
3-connected planar graph, it holds that: a) after coloring all pairs with their
2-WL color, the graph has fixing number 1 with respect to 1-WL, or b) there is
a 2-WL-definable matching that can be used to transform the graph into a
smaller one, or c) 2-WL detects a connected subgraph that is essentially the
graph of a Platonic or Archimedean solid, a prism, a cycle, or a bipartite
graph K_{2,\ell}. In particular, the graphs from case (a) are identified by
2-WL.
