In the 90's Clark, Colbourn and Johnson wrote a seminal paper, where they proved that maximum clique can be solved in polynomial time in unit disk graphs. Since then, the complexity of maximum clique in intersection graphs of (unit) d-dimensional balls has been investigated. For ball graphs, the problem is NP-hard, as shown by Bonamy et al. (FOCS '18). They also gave an efficient polynomial time approximation scheme (EPTAS) for disk graphs, however the complexity of maximum clique in this setting remains unknown. In this paper, we show the existence of a polynomial time algorithm for solving maximum clique in a geometric superclass of unit disk graphs. Moreover, we give partial results toward obtaining an EPTAS for intersection graphs of convex pseudo-disks.

中文翻译：

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计算圆盘图几何超类中的最大团

在 90 年代 Clark、Colbourn 和 Johnson 写了一篇开创性的论文，他们证明了最大团可以在单位圆盘图中的多项式时间内求解。从那时起，研究了（单位）d维球的交叉图中的最大集团的复杂性。对于球图，问题是 NP-hard，如 Bonamy 等人所示。(FOCS '18)。他们还为磁盘图提供了一种有效的多项式时间近似方案 (EPTAS)，但是这种设置中最大集团的复杂性仍然未知。在本文中，我们展示了在单位圆盘图的几何超类中求解最大团的多项式时间算法的存在。此外，我们给出了获得凸伪盘相交图的 EPTAS 的部分结果。