In the area of beyond-planar graphs, i.e. graphs that can be drawn with some local restrictions on the edge crossings, the recognition problem is prominent next to the density question for the different graph classes. For 1-planar graphs, the recognition problem has been settled, namely it is NP-complete for the general case, while optimal 1-planar graphs, i.e. those with maximum density, can be recognized in linear time. For 2-planar graphs, the picture is less complete. As expected, the recognition problem has been found to be NP-complete in general. In this paper, we consider the recognition of simple optimal 2-planar graphs. We exploit a combinatorial characterization of such graphs and present a linear time algorithm for recognition and embedding.

中文翻译：

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识别和嵌入简单的最优 2 平面图

在超平面图领域，即可以在边缘交叉处有一些局部限制的图，识别问题在不同图类的密度问题旁边是突出的。对于1-平面图，识别问题已经解决，即一般情况下它是NP-完全的，而最优1-平面图，即具有最大密度的那些，可以在线性时间内被识别。对于 2 平面图，图片不太完整。正如预期的那样，已经发现识别问题通常是 NP 完全的。在本文中，我们考虑对简单最优2-平面图的识别。我们利用这些图的组合特征，并提出了一种用于识别和嵌入的线性时间算法。