Adding One Edge to Planar Graphs Makes Crossing Number and 1-Planarity Hard
Adding One Edge to Planar Graphs Makes Crossing Number and 1-Planarity Hard
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DOI：
10.1137/120872310
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发表时间：
2012-03
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影响因子：
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通讯作者：
Sergio Cabello;B. Mohar
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文献类型：
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作者：
Sergio Cabello;B. Mohar
如果一个图可以通过添加一条边从平面图中获得，则该图是近平面图的。我们展示了一个令人惊讶的事实，即计算近平面图的交叉数是 NP 困难的。如果图的每条边最多与另一条边交叉，则该图是 1-平面的。我们证明，确定给定的近平面图是否是 1 平面是 NP 困难的。两种简化的主要思想是考虑在圆盘内同时绘制两个平面图的问题，其中一些顶点固定在圆盘的边界处。这就引出了锚定嵌入的概念，这是具有独立意义的。一个有趣的结果是，即使仅限于三次图，我们也获得了交叉数问题 NP 完整性的新几何证明。这就解决了 Hlin\v{e}n\'y 的问题。
A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This leads to the concept of anchored embedding, which is of independent interest. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hlin\v{e}n\'y.