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] 已证明这两个问题均为 NP-难问题；然而，他们在证明结论时需要大量高次顶点（针对第一个问题）或大量锚点（针对第二个问题）。令人惊讶的是，正如我们在本文中所证明的，仅需三个度大于 3 的顶点，或仅需三个锚点，就足以维持这些问题的困难性。这一新结果也改进了 [Hlin\v{e}n\'y 和 Salazar, 2015] 关于曲面上联合交叉数困难性的先前结论。在锚定情形下，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；在几乎平面情形下，我们的结果接近最优，因为最大度为 3 的几乎平面图的交叉数可以高效计算 [Riskin 1996, Cabello 和 Mohar 2011]。