Selecting a good representation of a solution-space is vital to solving any search and optimization problem. In particular, once regions of high performance are found, having the property that small changes in the candidate solution correspond to searching nearby neighborhoods provides the ability to perform effective local optimization. To achieve this, it is common for stochastic search algorithms, such as stochastic hillclimbing, evolutionary algorithms (including genetic algorithms), and simulated annealing, to employ Gray Codes for encoding ordinal points or discretized real numbers. In this paper, we present a novel method to label similar and/or close points within arbitrary graphs with small Hamming distances. The resultant point labels can be seen as an approximate high-dimensional variant of Gray Codes with standard Gray Codes as a subset of the labels found here. The labeling procedure is applicable to any task in which the solution requires the search algorithm to select a small subset of items out of many. Such tasks include vertex selection in graphs, knapsack-constrained item selection, bin packing, prototype selection for machine learning, and numerous scheduling problems, to name a few.