We study the two-player communication problem of determining whether two vertices $x, y$ are nearby in a graph $G$, with the goal of determining the graph structures that allow the problem to be solved with a constant-cost randomized protocol. Equivalently, we consider the problem of assigning constant-size random labels (sketches) to the vertices of a graph, which allow adjacency, exact distance thresholds, or approximate distance thresholds to be computed with high probability from the labels. Our main results are that, for monotone classes of graphs: constant-size adjacency sketches exist if and only if the class has bounded arboricity; constant-size sketches for exact distance thresholds exist if and only if the class has bounded expansion; constant-size approximate distance threshold (ADT) sketches imply that the class has bounded expansion; any class of constant expansion (i.e. any proper minor closed class) has constant-size ADT sketches; and a class may have arbitrarily small expansion without admitting constant-size ADT sketches.

翻译：我们研究两个玩家通信问题：在给定图 \(G\) 中判定两顶点 \(x, y\) 是否邻近。目标在于确定允许以常代价随机化协议解决该问题的图结构。等价地，我们考虑为图的顶点分配常大小随机标签（草图）的问题，该标签使得可从标签中以高概率计算邻接性、精确距离阈值或近似距离阈值。我们的主要结论是：对于单调图类，常大小邻接草图存在当且仅当该类具有有界树度；常大小精确距离阈值草图存在当且仅当该类具有有界扩展；常大小近似距离阈值（ADT）草图蕴含该类具有有界扩展；任意常扩展类（即任意真子式封闭类）均具有常大小ADT草图；但某类可能具有任意小的扩展而仍不允许常大小ADT草图存在。