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Graphs isomorphic to their interchanges are studied. Using prior results of more special cases, plus one new concept, it is possible to characterize all locally finite self-interchange graphs, finite and infinite, connected and disconnected, with loops and parallel edges admitted. All solutions are shown to be componentunions of graphs from six easily described classes.doi:10.2307/2037249 fatcat:xjzt4nmhnfcpdcv2jyqkpytny4