Due to data compression or low resolution, nearby vertices and edges of a graph drawn in the plane may be bundled to a common node or arc. We model such a “compromised” drawing by a piecewise linear map \(\varphi :G\rightarrow {\mathbb {R}}^2\). We wish to perturb \(\varphi \) by an arbitrarily small \(\varepsilon >0\) into a proper drawing (in which the vertices are distinct points, any two edges intersect in finitely many points, and no three edges have a common interior point) that minimizes the number of crossings. An \(\varepsilon \)-perturbation, for every \(\varepsilon >0\), is given by a piecewise linear map \(\psi _\varepsilon :G\rightarrow {\mathbb {R}}^2\) with \(\Vert \varphi -\psi _\varepsilon \Vert <\varepsilon \), where \(\Vert .\Vert \) is the uniform norm (i.e., \(\sup \) norm). We present a polynomial-time solution for this optimization problem when G is a cycle and the map \(\varphi \) has no spurs (i.e., no two adjacent edges are mapped to overlapping arcs). We also show that the problem becomes NP-complete (i) when G is an arbitrary graph and \(\varphi \) has no spurs, and (ii) when \(\varphi \) may have spurs and G is a cycle or a union of disjoint paths.

中文翻译：

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扰动图中的交叉最小化

由于数据压缩或分辨率较低，在平面中绘制的图形的附近顶点和边缘可能会捆绑到一个公共节点或弧上。我们通过分段线性映射\（\ varphi：G \ rightarrow {\ mathbb {R}} ^ 2 \）对这种“折衷”的图形进行建模。我们希望通过任意小的\（\ varepsilon> 0 \）来扰动\（\ varphi \）到适当的图形中（在该图形中，顶点是不同的点，任意两个边在有限的多个点处相交，并且没有三个边具有共同的内部点），以最大程度地减少穿越次数。每个\（\ varepsilon> 0 \）的\（\ varepsilon \）-摄动由分段线性映射\（\ psi _ \ varepsilon：G \ rightarrow {\ mathbb {R}} ^ 2 \）给出与\（\ Vert \ varphi-\ psi _ \ varepsilon \ Vert <\ varepsilon \），其中\（\ Vert。\ Vert \）是统一规范（即\（\ sup \）规范）。当G是一个循环且映射\（\ varphi \）没有杂散（即，没有两个相邻的边映射到重叠的弧）时，我们为该优化问题提供多项式时间解。我们还表明，当（i）G是任意图并且\（\ varphi \）没有杂散时，以及（ii）当\（\ varphi \）可能具有杂散且G是周期或不相交的路径的结合。