  Dynamic Complexity (as introduced by Patnaik and Immerman) tries to express
how hard it is to update the solution to a problem when the input is changed
slightly. It considers the changes required to some stored data structure
(possibly a massive database) as small quantities of data (or a tuple) are
inserted or deleted from the database (or a structure over some vocabulary).
The main difference from previous notions of dynamic complexity is that instead
of treating the update quantitatively by finding the the time/space trade-offs,
it tries to consider the update qualitatively, by finding the complexity class
in which the update can be expressed (or made). In this setting, DynFO, or
Dynamic First-Order, is one of the smallest and the most natural complexity
class (since SQL queries can be expressed in First-Order Logic), and contains
those problems whose solutions (or the stored data structure from which the
solution can be found) can be updated in First-Order Logic when the data
structure undergoes small changes.
  Etessami considered the problem of isomorphism in the dynamic setting, and
showed that Tree Isomorphism can be decided in DynFO. In this work, we show
that isomorphism of Planar 3-connected graphs can be decided in DynFO+ (which
is DynFO with some polynomial precomputation). We maintain a canonical
description of 3-connected Planar graphs by maintaining a database which is
accessed and modified by First-Order queries when edges are added to or deleted
from the graph. We specifically exploit the ideas of Breadth-First Search and
Canonical Breadth-First Search to prove the results. We also introduce a novel
method for canonizing a 3-connected planar graph in First-Order Logic from
Canonical Breadth-First Search Trees.
