Log space algorithms for modular decomposition tree Can we have log space algorithms for modular decomposition tree (see definition) for any graph?

If not, can we have log space algorithms for modular decomposition tree for any particular graph class?
If yes, then I would appreciate a pointer to the graph class or a reference.

Finally, can we have parallel algorithms for modular decomposition tree?
There is a logarithmic-space algorithm for the modular decomposition of any graph.

The result is part of the paper "Capturing Polynomial Time using Modular Decomposition", which will appear at LICS 2017 (http://lics.rwth-aachen.de/lics17/).
It is also part of the PhD thesis of the author. A preliminary version can be found here:
https://www2.informatik.hu-berlin.de/~grussien/grussien-pre-phd-thesis.pdf.

The algorithm uses the connection between modules and transitive orientation that was first shown by Gallai in 1967 and combines this with Reingold's logspace algorithm for undirected reachability. Given two vertices u and v, it can compute the minimal module containing both vertices. (A naive implementation of this closure would need some form of directed reachability.) 
Every module of the modular decomposition tree is equal to the union of certain modules that are minimal containing two vertices.