Open problems related to Graph isomorphism Presently I am doing literature survey on Graph isomorphism (GI) problem.

I would like to know some open questions related to the following  


What are the graph parameters for which fixed parameter tractability of GI is an open problem. 
What are the graph parameters, by fixing them polynomial time solvability of GI is not known.
Complexity of GI when restricted to many graph classes is equivalent to general GI (GI-Completeness). What are the graph classes for which GI-completeness is not known. 


Thank you.
For the first question: Graph Isomorphism has been considered for at least the following parameters for which fixed-parameter tractability is still open.


pathwidth / treewidth (see [2], has been asked here), maybe solved: http://arxiv.org/abs/1404.0818
cutwidth / bandwidth [1]
treewidth-k vertex deletion set size (feedback vertex set number in [7])
tree / path distance width (see [1]), connected tree distance width (see [3], however you can get quite close to the last one, see section 6.4. of my diploma thesis ) : solved by Y. Otachi  and P. Schweitzer : http://arxiv.org/abs/1403.7238
clique-width / shrub-depth (or SC-depth) (see [4])
maximum degree [5]
genus [6] / crossing number [8]


Note that there is active ongoing research for some them.

[1] : K. Yamazaki, H. L. Bodlaender, B. de Fluiter and D. M. Thilikos. Isomorphism
for graphs of bounded distance width. Algorithmica 24.2 (1999)

[2] : H. L. Bodlaender. Polynomial algorithms for graph isomorphism and chro-
matic index on partialk-trees. Journal of Algorithms 11.4 (1990)

[3] : Y. Otachi. Isomorphism for Graphs of Bounded Connected-Path-Distance-
Width. Algorithms and Computation. Springer, 2012

[4] : http://www.fi.muni.cz/~hlineny/res-en.html#recent

[5] : L. Babai and E. M. Luks. Canonical labeling of graphs. STOC '83.

[6] : I. S. Filotti and J. N. Mayer. A polynomial-time algorithm for determining the
isomorphism of graphs of fixed genus. STOC '80 / 
G. Miller. Isomorphism testing for graphs of bounded genus. STOC '80 

[7] : S. Kratsch and P. Schweitzer. Isomorphism for graphs of bounded feedback vertex set number. SWAT 2010 

[8] : http://math.mit.edu/news/summer/SPURprojects/2012Velednitsky.pdf