# 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 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. 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. 2.

A graphG=(V, E) has a perfectP k -matching if and only if for anyXV the number ofP k -critical components inG[V - X] is not greater than |X|.

3. 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.

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## References

1. [1]

I. Anderson, Perfect Matchings of a Graph,Journal of Combinatorial Theory B 10 (1971), 183–186.

2. [2]

G. Cornuéjols andW. Pulleyblank, A Matching Problem with Side Conditions,Discrete Mathematics 29 (1980), 135–159.

3. [3]

G. Cornuéjols andW. Pulleyblank, Perfect Triangle-Free 2-matchings,Mathematical Programming Study 13 (1980), 1–7.

4. [4]

G. Cornuéjols andW. Pulleyblank, The Travelling Salesman Polytope and {0,2}-matchings, inAnnals of Discrete Mathematics 16 (1982), 27–55.

5. [5]

J. Edmonds, Paths, Trees and Flowers,Canadian Journal of Mathematics 17 (1965), 449–467.

6. [6]

J. Edmonds, Maximum Matching and a Polyhedron with 0,1 Vertices,Journal of Research of the National Bureau of Standards 69 B (1965), 125–130.

7. [7]

L. Lovász, A Note on Factor-Critical Graphs,Studia Scientiarum Mathematicarum Hungarica 7 (1972), 279–280.

8. [8]

L. Lovász,Combinatorial Problems and Exercises, North Holland, 1979.

9. [9]

L. Lovász,Private Communication.

10. [10]

W. Pulleyblank,Faces of Matching Polyhedra, Ph. D. Thesis, University of Waterloo (1973).

11. [11]

W. Pulleyblank, Minimum Node Covers and 2-Bicritical Graphs,Mathematical Programming 17 (1979), 91–103.

12. [12]

W. Pulleyblank andJ. Edmonds, Facets of 1-matching Polyhedra, inHypergraph Seminar, (eds. C. Berge and D. K. Ray—Chaudhuri), Springer Verlag (1974), 214–242.

13. [13]

W. T. Tutte, The Factorization of Linear Graphs,Journal of the London Mathematical Society 22 (1947), 107–111.

14. [14]

W. T. Tutte, The Factors of Graphs,Canadian Journal of Mathematics 4 (1952), 314–328.

## Author information

### Affiliations

Authors

Dedicated to Tibor Gallai on his seventieth birthday

Supported by NSF grant ENG 79-02506.

Supported by Sonderforschungsbereich 21 (DFG), Institut für Operations Research, Universität Bonn. and by the National Science and Engineering Research Council of Canada.

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