We survey recent developments in the parameterized complexity of the graph isomorphism problem. Our focus lies on the fixed-parameter tractability of isomorphism testing for various graph parameters. Additionally, following Babai’s quasipolynomial time isomorphism test, there has been a series of results obtaining isomorphism algorithms running in time n polylog(k) for different graph parameters k.
We highlight the main technical ideas underlying these algorithms and state various open questions for future research.