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.

中文翻译：

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将图嵌入到嵌入图中

如果可以通过任意小的扰动将图 G 转换为嵌入，则平面中图 G 的（可能是退化的）绘图可通过嵌入近似。我们表明，如果所需的 G 嵌入属于固定同位素类，则可以在多项式时间内测试平面中平面图 G 的分段线性绘制是否可通过嵌入逼近。换句话说，我们证明了带有嵌入式管道的 c 平面性对于具有指定组合嵌入的图形是易于处理的。据我们所知，以前基本上只有当 G 是循环时才知道类似的结果。