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

- Received:

DOI: 10.1007/BF02579340

- Cite this article as:
- Cornuéjols, G. & Pulleyblank, W.R. Combinatorica (1983) 3: 35. doi:10.1007/BF02579340

## Abstract

*(perfect)*2-

*matching*in a graph

*G=(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 in

*G*, is an example of a perfect 2-matching. For

*k*satisfying 1≦

*k*≦|

*V*|, we let

*P*

_{k}denote the problem of finding a perfect 2-matching of

*G*such that any cycle in the solution contains more than

*k*edges. We call such a matching a

*perfect P*

_{k}-

*matching*. Then for

*k*<

*l*, the problem

*P*

_{k}is a relaxation of

*P*

_{1}. Moreover if |

*V*| is odd, then

*P*

_{1V1–2}is simply the problem of determining whether or not

*G*is Hamiltonian. A graph is

*P*

_{k}-

*critical*if it has no perfect

*P*

_{k}-matching but whenever any node is deleted the resulting graph does have one. If

*k*=|

*V*|, then a graph

*G*=(

*V, E*) is

*P*

_{k}-critical if and only if it is

*hypomatchable*(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 is

*P*_{k}-critical, then it is also*P*_{l}-critical for all larger*l*. In particular, for all*k, P*_{k}-critical graphs are hypomatchable. - 2.
A graph

*G*=(*V, E*) has a perfect*P*_{k}-matching if and only if for any*X*⊆*V*the number of*P*_{k}-critical components in*G[V - X]*is not greater than |*X*|. - 3.
The problem

*P*_{k}can be solved in polynomial time provided we can recognize*P*_{k}-critical graphs in polynomial time. In addition, we describe a procedure for recognizing*P*_{k}-critical graphs which is polynomial in the size of the graph and exponential in*k*.