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 $\Omega(n)$. 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 $\Omega(n^{1-\delta})$ for any $\delta>0$ 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：局部验证的目标是本地化地使图$G$中的顶点确信$G$满足给定性质。证明者向图的顶点分配短证书，随后顶点允许检查自身及邻居的证书，并仅基于此局部视角判断$G$是否满足该性质。若图确实满足该性质，所有顶点必须接受该实例；否则（对于任何可能的证书分配），至少有一个顶点必须拒绝该实例。目标是尽可能减小证书的尺寸。本文研究了几何与拓扑图类的局部验证问题。已知在$n$顶点图中，平面性可通过大小为$O(\log n)$的证书局部验证，但我们证明多个密切相关的图类需要大小为$\Omega(n)$的证书，包括便士图、单位距离图、方格网格的（诱导）子图、1-平面图及单位正方形图。这些界在常数因子内是紧的，且首次提供了证书需线性大小的遗传（甚至单调）图类实例。对于单位圆盘图，我们得到证书大小的下界$\Omega(n^{1-\delta})$（对任意$\delta>0$）与上界$O(n \log n)$。下界通过证明所考虑图的刚性性质获得，这一结果可能具有独立意义。