The goal of local certification is to locally convince the vertices of a graph $G$ that $G$ satisfies a given property. A prover assigns short certificates to the vertices of the graph, then the vertices are allowed to check their certificates and the certificates of their neighbors, and based only on this local view, they must decide whether $G$ satisfies the given property. If the graph indeed satisfies the property, all vertices must accept the instance, and otherwise at least one vertex must reject the instance (for any possible assignment of certificates). The goal is to minimize the size of the certificates. In this paper we study the local certification of geometric and topological graph classes. While it is known that in $n$-vertex graphs, planarity can be certified locally with certificates of size $O(\log n)$, we show that several closely related graph classes require certificates of size $Ω(n)$. This includes penny graphs, unit-distance graphs, (induced) subgraphs of the square grid, 1-planar graphs, and unit-square graphs. These bounds are tight up to a constant factor and give the first known examples of hereditary (and even monotone) graph classes for which the certificates must have linear size. For unit-disk graphs we obtain a lower bound of $Ω(n^{1-δ})$ for any $δ>0$ on the size of the certificates, and an upper bound of $O(n \log n)$. The lower bounds are obtained by proving rigidity properties of the considered graphs, which might be of independent interest.

翻译：局部验证的目标是让图$G$的顶点在局部确信$G$满足给定性质。证明者为图的顶点分配短证书，随后顶点被允许检查自身及其邻居的证书，仅基于这一局部视图，它们必须判定$G$是否满足给定性质。若图确实满足该性质，则所有顶点必须接受该实例；否则至少存在一个顶点必须拒绝实例（对任意可能的证书分配方案）。研究目标在于最小化证书尺寸。本文研究几何与拓扑图类的局部验证问题。虽然已知在$n$顶点图中，平面性可通过尺寸为$O(\log n)$的证书进行局部验证，但我们证明若干密切相关的图类需要$Ω(n)$尺寸的证书。这包括便士图、单位距离图、方形网格的（诱导）子图、1-平面图以及单位方形图。这些界限在常数因子内是紧的，并首次给出了需要线性尺寸证书的遗传性（甚至单调性）图类实例。对于单位圆盘图，我们获得证书尺寸的下界为$Ω(n^{1-δ})$（对任意$δ>0$），上界为$O(n \log n)$。下界通过证明所研究图的刚性性质获得，该性质可能具有独立的研究价值。