Embedding Graphs into Embedded Graphs

A (possibly degenerate) drawing of a graph G in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing whether a piece-wise linear drawing of a planar graph G in the plane is approximable by an embedding can be carried out in polynomial time, if a desired embedding of G belongs to a fixed isotopy class. In other words, we show that c-planarity with embedded pipes is tractable for graphs with prescribed combinatorial embedding. To the best of our knowledge, an analogous result was previously known essentially only when G is a cycle.