Before revision:  We show that it is NP-complete to decide whether a given edge can be inserted into a simple drawing , by this solving an open question by Arroyo, Derka, and Parada and a pseudosegment \sigma, it can be decided in polynomial time whether there exists a pseudocircle \Phi_\sigma extending \sigma for which \mathcal{A}\cup\{\Phi_\sigma\} is again an arrangement of pseudocircles} .
After revision:  We show that it is NP-complete to decide whether a given edge can be inserted into a simple drawing . This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles \mathcal{A and a pseudosegment \sigma, it can be decided in polynomial time whether there exists a pseudocircle \Phi_\sigma extending \sigma for which \mathcal{A}\cup\{\Phi_\sigma\} is again an arrangement of pseudocircles} .