Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to logspace reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in logspace [Datta, Limaye, Nimbhorkar 08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell's algorithm for tree canonization, along with changes in the space complexity analysis.
The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and
a group theoretic lemma to bound the number of automorphisms of a
colored 3-connected planar graph. This lemma is crucial for the
reduction to work in logspace.