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 d-dimensional (unit) 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 a geometric superclass of unit disk graphs. Moreover, we give partial results toward obtaining an EPTAS for intersection graphs of convex pseudo-disks.

20 世纪 90 年代，Clark、Colbourn 和 Johnson 撰写了一篇开创性的论文，证明了单位圆盘图中的最大团可以在多项式时间内求解。从那时起，人们开始研究d维（单位）球的交集图中最大团的复杂性。对于球图，问题是 NP 困难的，如 Bonamy 等人所示。 （FOCS '18）。他们还给出了磁盘图的有效多项式时间近似方案（EPTAS）。然而，这种情况下最大派系的复杂性仍然未知。在本文中，我们展示了单位圆盘图的几何超类的多项式时间算法的存在性。此外，我们给出了获得凸伪盘相交图的 EPTAS 的部分结果。