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

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 letP _k 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 P _k -matching. Then fork<l, the problemP _k is a relaxation ofP ₁. Moreover if |V| is odd, thenP _1V1–2 is simply the problem of determining whether or notG is Hamiltonian. A graph isP _k -critical if it has no perfectP _k -matching but whenever any node is deleted the resulting graph does have one. Ifk=|V|, then a graphG=(V, E) isP _k -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 isP _k -critical, then it is alsoP _l -critical for all largerl. In particular, for allk, P _k -critical graphs are hypomatchable.
2. A graphG=(V, E) has a perfectP _k -matching if and only if for anyX?V the number ofP _k -critical components inGV - X] is not greater than |X|.
3. The problemP _k can be solved in polynomial time provided we can recognizeP _k -critical graphs in polynomial time. In addition, we describe a procedure for recognizingP _k -critical graphs which is polynomial in the size of the graph and exponential ink.