Inherent Symmetries of graphs, paths, and Traveling Salesperson Problems

TL;DR

This paper reveals that even asymmetric graphs possess inherent symmetry structures that determine all closed path properties, by decomposing graphs into core components and superfluous parts, applicable to various TSP variants.

Contribution

It introduces a decomposition method to identify inherent symmetry structures in graphs, extending to incomplete and asymmetric cases, advancing understanding of TSP problem properties.

Findings

Symmetries in graphs are revealed through decomposition.

Different symmetry types are identified for various TSP settings.

The method applies to incomplete and asymmetric graphs.

Abstract

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.