A simple drawing {\em  D(G) of a graph G = (V,E) is a drawing in which two edges have at most one point in common that is either an endpoint or a proper crossing. An edge e from the complement of G can be inserted {\em  into D(G) if there exists a simple drawing of G ' = (V, E\cup \{e \ D(G) as a subdrawing. 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} .
A {\em simple drawing D(G) of a graph G is one where each pair of edges share at most one point : either a common endpoint or a proper crossing. An edge e in the complement of G can be {\em inserted into D(G) if there exists a simple drawing of G +e extending D(G) . As a result of Levi's Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP -complete to decide whether one 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} .
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simple drawing
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from
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inserted
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inserted
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+e extending
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as a subdrawing. We
. As a result of Levi's Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we
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NP-complete
NP -complete
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a given
one
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, by this solving an open question by Arroyo, Derka, and Parada
. 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