Inserting a prescribed set of edges into a simple drawing is NP-complete, the maximization version is APX-hard, and one-edge insertion is polynomial when the endpoints form a dominating set.
Information Processing Letters 110(12-13), 521–523 (2010)
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Extending Simple Drawings
Inserting a prescribed set of edges into a simple drawing is NP-complete, the maximization version is APX-hard, and one-edge insertion is polynomial when the endpoints form a dominating set.