Locally checkable proofs for graph properties were introduced by G\"o\"os and Suomela \cite{GS}. Roughly speaking, a graph property $\cP$ is locally checkable in constant-time, if the vertices of a graph having the property can be convinced, in a short period of time not depending on the size of the graph, that they are indeed vertices of a graph having the given property. For a given $\eps>0$, we call a property $\cP$ $\eps$-locally checkable in constant-time if the vertices of a graph having the given property can be convinced at least that they are in a graph $\eps$-close to the given property. We say that a property $\cP$ is almost locally checkable in constant-time, if for all $\eps>0$, $\cP$ is $\eps$-locally checkable in constant-time. It is not hard to see that in the universe of bounded degree graphs planarity is not locally checkable in constant-time. However, the main result of this paper is that planarity of bounded degree graphs is almost locally checkable in constant-time. The proof is based on the surprising fact that although graphs cannot be convinced by their planarity or hyperfiniteness, planar graphs can be convinced by their own hyperfiniteness. The reason behind this fact is that the class of planar graphs are not only hyperfinite but possesses Property A of Yu.

中文翻译：

---

平面度（几乎）可以在恒定时间内进行局部检查

G\"o\"os 和 Suomela \cite{GS} 引入了图属性的本地可检查证明。粗略地说，一个图属性 $\cP$ 在恒定时间内是局部可检查的，如果可以在不依赖于图的大小的短时间内确信具有该属性的图的顶点，它们确实是具有给定属性的图的顶点。对于给定的 $\eps>0$，我们称属性 $\cP$ $\eps$-在恒定时间内可局部检查，如果具有给定属性的图的顶点至少可以确信它们在图中$\eps$-接近给定的属性。我们说属性 $\cP$ 在恒定时间内几乎是局部可检查的，如果对于所有 $\eps>0$，$\cP$ 是 $\eps$-在恒定时间内可局部检查的。不难看出，在有界度图的宇宙中，平面性在恒定时间内是不可局部检查的。然而，本文的主要结果是有界度图的平面性在恒定时间内几乎是局部可检查的。证明基于一个令人惊讶的事实，即虽然图不能被它们的平面性或超有限性所说服，但平面图可以被它们自己的超有限性所说服。这个事实背后的原因是平面图的类不仅是超有限的，而且具有于的性质 A。平面图可以被它们自己的超有限性所说服。这个事实背后的原因是平面图的类不仅是超有限的，而且具有于的性质 A。平面图可以被它们自己的超有限性所说服。这个事实背后的原因是平面图的类不仅是超有限的，而且具有于的性质 A。