On the Structure of Unique Shortest Paths in Graphs

This paper develops a structural theory of unique shortest paths in real-weighted graphs. Our main goal is to characterize exactly which sets of node sequences, which we call path systems, can be realized as unique shortest paths in a graph with arbitrary real edge weights. We say that such a path system is strongly metrizable. An easy fact implicit in the literature is that a strongly metrizable path system must be consistent, meaning that no two of its paths may intersect, split apart, and then intersect again. Our main result characterizes strong metrizability via some new forbidden intersection patterns along these lines. In other words, we describe a family of forbidden patterns beyond consistency, and we prove that a path system is strongly metrizable if and only if it is consistent and it avoids all of the patterns in this family. We offer separate (but closely related) characterizations in this way for the settings of directed, undirected, and directed acyclic graphs. Our characterizations are based on a new connection between shortest paths and topology; in particular, our new forbidden patterns are in natural correspondence with two-colored topological $2$-manifolds, which we visualize as polyhedra. We believe that this connection may be of independent interest, and we further show that it implies some additional structural corollaries that seem to suggest new and possibly deep-rooted connections between these areas.