NP-hard problems on cographs This question is similar to NP-hard problems on trees:

There is a large number of NP-complete problems that are tractable on cographs. Are there any known problems that remain NP-complete when restricted to cographs?

To be more precise I am interested in examples where the input consists solely of an undirected, unweighted cograph.

Two remarks:


For weighted cographs such a problem is mentioned here - TSP with two travellers
Cographs are the "base class" of clique-width such as trees are the base class for tree-width.


UPDATE

Some further thoughts (I am not quite sure about):
If the input is really just a cograph, the question has to be of the sort "Does the cograph have property X?". It would be enough if such a problem existed for trees, since then the question could be "Does the cotree of the cograph have property X?".
Perhaps my favorite open problem is of interest: the edge clique-cover problem on cographs. In the edge clique-cover problem, you want to cover the edges of the cograph with a minimal number of cliques. It is unknown if this problem is NP-complete.

To illustrate that the problem is probably hard, let $K_n^m$ be the complete multipartite graph with $m$ partite sets each of size $n$. This is a cograph. There exist $m - 2$ pairwise orthogonal Latin squares of order $n$ if and only if the edge clique-cover of $K_n^m$ is $n^2$. This was shown by Park, Kim and 
 Sano. This is a formula for the "cocktail party graph", that is, the case where $n = 2$.