Polynomial problems in graph classes defined by forbidden induced cyclic subgraphs Crossposted from MO.

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 NP-hard 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

Edit

Negative for IS and Domination is in this paper. Page 2, the graphs $S_{i,j,k}$.
I think there is a number of hard problems that become easy for triangle-free graphs; especially those deal directly with triangles such as Partition Into Triangles (Does G have a partition into triangles?). Other less trivial examples include:


Stable Cutset Problem (Does G have an independent set S such that G-S is disconnected?). See:  On stable sutsets in graphs, Discrete Applied Math. 105 (2000) 39-50.  
Intersection Graph Basis (Is G the intersection graph of subsets of a k-element ground set?). See: Problem [GT59] in: Garey &amp; Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness.