  Without imposing restrictions on a weighted graph's arc lengths, symmetry
structures cannot be expected. But, they exist. To find them, the graphs are
decomposed into a component that dictates all closed path properties (e.g.,
shortest and longest paths), and a superfluous component that can be removed.
The simpler remaining graph exposes inherent symmetry structures that form the
basis for all closed path properties. For certain asymmetric problems, the
symmetry is that of three-cycles; for the general undirected setting it is a
type of four-cycles; for general directed problems with asymmetric costs, it is
a product of three and four cycles. Everything extends immediately to
incomplete graphs.
