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.
When parameterized by its Grundy number, it can be calculated in fixed-parameter tractable time for chordal graphs and claw-free graphs, and also (utilizing general results on subgraph isomorphism in sparse graphs to search for atoms) for graphs of bounded expansion.