Is clique-width preserved under edge contractions?

Let $G$ be class of graphs with bounded clique-width. In each graph in $G$ some edges are contracted (e.g. randomly). Is now the clique-width still bounded?

In case it is (in general) no longer bounded, I would be very interested in a counter-example.

This may be a pre-answer: in this 2007 arXiv paper (Problem 4.9), it is stated as an open problem whether one can find a graph $G$ and an edge $\{x,y\} \in E(G)$ such that $cw(G) < cw(G^{x,y})$, where $G^{x,y}$ is the graph $G$ with edge $\{x,y\}$ contracted.

• Thanks a lot for your answer! That is a valuable reference for me. In case no one has solved it in the meantime, my question is more or less answered :) – Martin Lackner Dec 22 '10 at 18:37
• Isn’t that problem the opposite direction from what is being asked here? – Tsuyoshi Ito Dec 23 '10 at 10:53
• Only in the sense that this question asks for a counterexample. – Martin Lackner Dec 23 '10 at 17:31

This recent paper finally proves that edge contractions do not preserve the property that a set of graphs has bounded clique-width.