Our algorithm for K3,3 minor-free graphs proceeds by decomposition into triconnected components, which are known to be either planar or K5 components [Vazirani89]. This gives a triconnected component tree similar to that for planar graphs. An extension of the log-space algorithm of [DLNTW09] can then be used to decide the isomorphism problem.
For K5 minor-free graphs, it is known from [Khuller88] that the 4-connected components are either planar graphs or a four-rung mobius ladder on 8 vertices. We give an algorithm to get a unique
decomposition of K5 minor-free graphs into triconnected and four-connected planar components, and construct triconnected and four-connected component trees. As the algorithm of [DLNTW09] does not deal with four-connected component trees, it needs to be modified in a quite non-trivial way.