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/']
Here are some additional examples to Mon Tag's answer :

* The Disconnected Cutset problem (Does $G$ admit a set of vertices $S$ such that $G-S$ and the subgraph of $G$ induced by $S$ are disconnected) is NP-complete (see [here](http://dx.doi.org/10.1007/978-3-642-23786-7_43)). It is easy to see that this problem is polynomially solvable for triangle-free graphs (hence also the Stable Cutset problem as mentioned by Mon Tag).
* Recognizing triangular line graphs is NP-complete (see [here](http://www.sciencedirect.com/science/article/pii/S0012365X11005449)), It is also 
easy to see that this problem becomes polynomially for triangle-free input graphs.
* Computing the maximum connected matching is hard (see [here](http://www.discuss.wmie.uz.zgora.pl/gt/index.php?doi=10.7151/dmgt.1206). 
A matching is connected if, for any pair of the matching edges, there is another edge of the graph incident to both of them). 
It can be proved that this problem is polynomially solvable for $(C\_3, C\_4, C\_5)$-free graphs.
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"