Let $C$ be a graph class defined by a finite number of forbidden induced subgraphs, all of which are cyclic (contain at least one cycle).
Are there graph problems that can be solved in polynomial time for $C$ other than Clique and Clique cover?
If I remember correctly, this is impossible for independent set (unless $P=NP$).
Search in graphclasses.org didn't find any.
A class for which Clique and Clique cover are polynomial is (C5,C6,X164,X165,sunlet4,triangle)-free