When parameterized by the Grundy number itself, it can be computed in fixed-parameter tractable time for chordal graphs and claw-free graphs, and also (using general results on subgraph isomorphism in sparse graphs to search for atoms)  for graphs of bounded expansion.
For chordal graphs and claw-free graphs, the parameterization by Grundy number allows for fixed-parameter tractable computation. Additionally, using results from subgraph isomorphism in sparse graphs to look for atoms, this method can be applied to graphs with bounded expansion.