Critical graphs, matchings and tours or a hierarchy of relaxations for the travelling salesman problem

Abstract

A(perfect) 2-matching in a graphG=(V, E) is an assignment of an integer 0, 1 or 2 to each edge of the graph in such a way that the sum over the edges incident with each node is at most (exactly) two. The incidence vector of a Hamiltonian cycle, if one exists inG, is an example of a perfect 2-matching. Fork satisfying 1≦k≦|V|, we letPk denote the problem of finding a perfect 2-matching ofG such that any cycle in the solution contains more thank edges. We call such a matching aperfect Pk-matching. Then fork<l, the problemPk is a relaxation ofP1. Moreover if |V| is odd, thenP1V1–2 is simply the problem of determining whether or notG is Hamiltonian. A graph isPk-critical if it has no perfectPk-matching but whenever any node is deleted the resulting graph does have one. Ifk=|V|, then a graphG=(V, E) isPk-critical if and only if it ishypomatchable (the graph has an odd number of nodes and whatever node is deleted the resulting graph has a perfect matching). We prove the following results: 1. If a graph isPk-critical, then it is alsoPl-critical for all largerl. In particular, for allk, Pk-critical graphs are hypomatchable. 2. A graphG=(V, E) has a perfectPk-matching if and only if for anyX⊆V the number ofPk-critical components inG[V - X] is not greater than |X|. 3. The problemPk can be solved in polynomial time provided we can recognizePk-critical graphs in polynomial time. In addition, we describe a procedure for recognizingPk-critical graphs which is polynomial in the size of the graph and exponential ink.