Crossposted from [MO](https://mathoverflow.net/questions/171493/polynomial-problems-in-graph-classes-defined-by-forbidden-induced-cyclic-subgrap).

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](http://graphclasses.org/classes/gc_1022.html)

**Edit**

Negative for IS and Domination is in [this paper](http://www.sciencedirect.com/science/article/pii/S0304397505008327). Page 2, the graphs $S\_{i,j,k}$.
['https://cstheory.stackexchange.com/questions/24876', 'https://cstheory.stackexchange.com', 'https://cstheory.stackexchange.com/users/22348/']
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 & Johnson, *Computers and Intractability: A Guide to the Theory of NP-Completeness.*
MAX-CUT remains NP-complete.

**Lemma 3.2** simple max-cut is NP-complete in the following two classes of graphs:

graphs not containing cycles of length at most $k$, for every $k \ge 3$.

They are subdividing an edge twice.

From "MAX-CUT and containment relations in graphs, Marcin Kaminski"