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：本文研究几乎平面图精确交叉数的计算问题，以及与之密切相关的平面图对精确锚定交叉数的计算问题。[Cabello and Mohar, 2013]已证明这两个问题均属于NP难问题；尽管他们需要无限多个高度数顶点（针对第一个问题）或无限多个锚点（针对第二个问题）来证明其结论。然而出人意料的是，如本文所证，仅需三个度数大于3的顶点，或仅需三个锚点，便足以保持这些问题的计算困难性。该新结果同时改进了[Hliněný and Salazar, 2015]关于曲面联合交叉数计算困难性的先前结论。在锚定情形下，我们的结果已达到最优边界，因为具有两个锚点的平面图对的锚定交叉数是平凡可解的；在几乎平面情形下，该结果亦接近最优边界，因为对于最大度数为3的几乎平面图，其交叉数可在多项式时间内计算[Riskin 1996, Cabello and Mohar 2011]。