This one has been bugging me for a while. A long time ago in undergrad, I noticed this while learning about TSP. Nobody recognized it and I basically gave up.
Given a hamiltonian path, any subpath will consist of a start, end, and set of interior vertices. Any ordering of the interior vertices will possibly yield a path existing within the graph, that when substituted for the original will yield another valid hamiltonian path. These subpaths would therefore form an equivalence class.
Under this relation, two paths would be equivalent if they have the same start vertex, the same end vertex, and the same set of interior vertices.
Does this equivalence class have a name? If so, are there any TSP optimization algorithms that make use of this?