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 to 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. For unit-disk graphs we obtain a lower bound of $\Omega(n^{1-\delta})$ for any $\delta>0$ on the size of the certificates. All our results are tight up to a $n^{o(1)}$ factor, and give the first known examples of hereditary (and even monotone) graph classes for which the certificates must have polynomial size. The lower bounds are obtained by proving rigidity properties of the considered graphs, which might be of independent interest.

翻译：局部认证的目标是让图$G$的顶点通过局部信息验证$G$是否满足给定性质。证明者向图的顶点分配短证书，随后各顶点允许检查自身及其邻居的证书，并仅基于这种局部视图判断$G$是否满足给定性质。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。目标是使证书的尺寸最小化。本文研究几何与拓扑图类的局部认证问题。已知在$n$顶点图中，平面性可通过大小为$O(\log n)$的证书进行局部认证，但我们证明若干密切相关图类要求证书尺寸为$\Omega(n)$，包括便士图、单位距离图、方格网格的（诱导）子图、1-平面图以及单位正方形图。对于单位圆盘图，我们得出对任意$\delta>0$，证书尺寸的下界为$\Omega(n^{1-\delta})$。所有结果在$n^{o(1)}$因子范围内都是紧的，并首次给出遗传（甚至单调）图类中证书需多项式尺寸的已知示例。下界通过证明所考虑图的刚性性质获得，该性质本身可能具有独立研究价值。