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：我们研究两玩家通信问题：判定图 $G$ 中两个顶点 $x, y$ 是否相邻近，旨在确定允许该问题通过常数代价随机化协议求解的图结构。等价地，我们考虑为图顶点分配常数大小随机标签（草图）的问题，使得从标签中能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，存在常数大小的邻接草图当且仅当该类有界树度；存在常数大小的精确距离阈值草图当且仅当该类有界扩张；存在常数大小的近似距离阈值（ADT）草图蕴含该类有界扩张；任何常数扩张类（即任何真亏格闭类）均存在常数大小的ADT草图；而一类图可能具有任意小的扩张却不允许常数大小的ADT草图。