Choosability on H-free graphs

A graph is H-free if it has no induced subgraph isomorphic to H. We determine the computational complexity of the Choosability problem restricted to H-free graphs for every graph H that does not belong to {K1,3,P1+P2,P1+P3,P4}{K1,3,P1+P2,P1+P3,P4}. We also show that if H is a linear forest, then the problem is fixed-parameter tractable when parameterized by k