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Embedding Graphs into Embedded Graphs
Algorithmica ( IF 0.9 ) Pub Date : 2020-06-11 , DOI: 10.1007/s00453-020-00725-3
Radoslav Fulek

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:

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。
更新日期:2020-06-11
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