As a recent breakthrough result of Babai ignites excitement about the graph isomorphism problem, the quest to resolve its complexity continues.
After a brief overview over the current landscape of results, I describe some decomposition techniques arising in particular in the structural theory of graphs
with a forbidden minor. General structure theorems yield canonical decompositions whose parts correspond to the maximal tangles.
These tangles may be viewed as describing "k-connected components". I will describe algorithmic versions of such theorems.
These algorithmic versions at hand and using the notion of treelike decompositions, it is sometimes possible to further decompose said parts in a canonical fashion.
Such canonical decompositions find their application in algorithms for the graph isomorphism problem. I will describe how they can be used to test isomorphism
of graphs of bounded rank width (or equivalently of bounded clique width) in polynomial time.