Encoding shortest paths in graphs assuming the code is queried using bit-wise comparison

One model of message delivery in a computer network is based on labelling each edge by a subset of a (reasonably small) universal set, and then encoding a path as the union of the labels of its edges. Earlier work suggested using random edge labels, and that approach has a disadvantage of producing errors (false positives). We demonstrate that if we make an assumption about the shape of the network (in this paper we consider networks with a dense core and a tree-like periphery) and assume that messages are delivered along shortest paths, we can label edges in a way which prevents any false positives.