Extending simple drawings with one edge is hard
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A simple drawing 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 into D(G) if there exists a simple drawing of G' = (V, E∪{e}) containing 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.
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