Critical graphs, matchings and tours or a hierarchy of relaxations for the travelling salesman problem
 G. Cornuéjols,
 W. R. Pulleyblank
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
A(perfect) 2matching 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 2matching. Fork satisfying 1≦k≦V, we letP _{ k } denote the problem of finding a perfect 2matching 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:

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.

A graphG=(V, E) has a perfectP _{ k }matching if and only if for anyX⊆V the number ofP _{ k }critical components inG[V  X] is not greater than X.

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.
 I. Anderson, Perfect Matchings of a Graph,Journal of Combinatorial Theory B 10 (1971), 183–186. CrossRef
 G. Cornuéjols andW. Pulleyblank, A Matching Problem with Side Conditions,Discrete Mathematics 29 (1980), 135–159. CrossRef
 G. Cornuéjols andW. Pulleyblank, Perfect TriangleFree 2matchings,Mathematical Programming Study 13 (1980), 1–7.
 G. Cornuéjols andW. Pulleyblank, The Travelling Salesman Polytope and {0,2}matchings, inAnnals of Discrete Mathematics 16 (1982), 27–55.
 J. Edmonds, Paths, Trees and Flowers,Canadian Journal of Mathematics 17 (1965), 449–467.
 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.
 L. Lovász, A Note on FactorCritical Graphs,Studia Scientiarum Mathematicarum Hungarica 7 (1972), 279–280.
 L. Lovász,Combinatorial Problems and Exercises, North Holland, 1979.
 L. Lovász,Private Communication.
 W. Pulleyblank,Faces of Matching Polyhedra, Ph. D. Thesis, University of Waterloo (1973).
 W. Pulleyblank, Minimum Node Covers and 2Bicritical Graphs,Mathematical Programming 17 (1979), 91–103. CrossRef
 W. Pulleyblank andJ. Edmonds, Facets of 1matching Polyhedra, inHypergraph Seminar, (eds. C. Berge and D. K. Ray—Chaudhuri), Springer Verlag (1974), 214–242.
 W. T. Tutte, The Factorization of Linear Graphs,Journal of the London Mathematical Society 22 (1947), 107–111. CrossRef
 W. T. Tutte, The Factors of Graphs,Canadian Journal of Mathematics 4 (1952), 314–328.
 Title
 Critical graphs, matchings and tours or a hierarchy of relaxations for the travelling salesman problem
 Journal

Combinatorica
Volume 3, Issue 1 , pp 3552
 Cover Date
 19830301
 DOI
 10.1007/BF02579340
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 05 C 38
 Industry Sectors
 Authors

 G. Cornuéjols ^{(1)}
 W. R. Pulleyblank ^{(2)}
 Author Affiliations

 1. G.S.I.A., CarnegieMellon University, 15213, Pittsburgh, PA, USA
 2. Department of Combinatorics and Optimization, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada