The adjacency relation on the traveling salesman polytope is NP-Complete

We consider the problem of determining whether two traveling salesman tours correspond to non-adjacent vertices of the convex polytope associated with the traveling salesman problem. This problem is shown to be NP-Complete for both the symmetric and nonsymmetric traveling salesman problem. Several implications are discussed.This Research was supported by NSF Grant GK-420488, the U.S. Army Research Office-Durham under Grant DAHC04-75-G0192, and an IBM Fellowship.