On the Structure of Unique Shortest Paths in Graphs
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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 appear 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 forbidden intersection patterns along these lines. In other words, we describe a new family of forbidden patterns beyond consistency, and we prove that a path system is strongly metrizable if and only if it consistent and it avoids all of these new patterns. 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 certain boundary operators used in homology, which is used to prove several additional structural corollaries and seems to suggest new and possibly deep-rooted connections between shortest paths and topology.
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