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

Article

- Received:

DOI: 10.1007/BF02579340

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

- 15 Citations
- 75 Downloads

## Abstract

A

*(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*.

### AMS subject classification (1980)

05 C 38## Copyright information

© Akadémiai Kiadó 1983