In this paper we deal with the problem of computing the exact crossing number of almost planar graphs and the closely related problem of computing the exact anchored crossing number of a pair of planar graphs. It was shown by [Cabello and Mohar, 2013] that both problems are NP-hard; although they required an unbounded number of high-degree vertices (in the first problem) or an unbounded number of anchors (in the second problem) to prove their result. Somehow surprisingly, only three vertices of degree greater than 3, or only three anchors, are sufficient to maintain hardness of these problems, as we prove here. The new result also improves the previous result on hardness of joint crossing number on surfaces by [Hlin\v{e}n\'y and Salazar, 2015]. Our result is best possible in the anchored case since the anchored crossing number of a pair of planar graphs with two anchors each is trivial, and close to being best possible in the almost planar case since the crossing number is efficiently computable for almost planar graphs of maximum degree 3 [Riskin 1996, Cabello and Mohar 2011].

翻译：本文研究了计算几乎平面图精确交叉数的问题，以及与之密切相关的计算一对平面图精确锚定交叉数的问题。Cabello 和 Mohar（2013）证明了这两个问题都是 NP 难的；尽管他们需要无界数量的高度顶点（在第一个问题中）或无界数量的锚点（在第二个问题中）来证明其结论。令人惊讶的是，正如我们在此证明的，仅需三个度数大于 3 的顶点，或仅需三个锚点，就足以维持这些问题的难度。这一新结果也改进了 Hliněný 和 Salazar（2015）关于曲面上联合交叉数难解性的先前结论。在锚定情况下，我们的结果是最优的，因为每对具有两个锚点的平面图的锚定交叉数是平凡的；在几乎平面图情况下，该结果接近最优，因为最大度数为 3 的几乎平面图的交叉数可以高效计算（Riskin 1996，Cabello and Mohar 2011）。