Combinatorica

, Volume 3, Issue 1, pp 35–52

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

  • G. Cornuéjols
  • W. R. Pulleyblank
Article

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 letPk 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 Pk-matching. Then fork<l, the problemPk is a relaxation ofP1. Moreover if |V| is odd, thenP1V1–2 is simply the problem of determining whether or notG is Hamiltonian. A graph isPk-critical if it has no perfectPk-matching but whenever any node is deleted the resulting graph does have one. Ifk=|V|, then a graphG=(V, E) isPk-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 isPk-critical, then it is alsoPl-critical for all largerl. In particular, for allk, Pk-critical graphs are hypomatchable.

     
  2. 2.

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

     
  3. 3.

    The problemPk can be solved in polynomial time provided we can recognizePk-critical graphs in polynomial time. In addition, we describe a procedure for recognizingPk-critical graphs which is polynomial in the size of the graph and exponential ink.

     

AMS subject classification (1980)

05 C 38 

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References

  1. [1]
    I. Anderson, Perfect Matchings of a Graph,Journal of Combinatorial Theory B 10 (1971), 183–186.CrossRefGoogle Scholar
  2. [2]
    G. Cornuéjols andW. Pulleyblank, A Matching Problem with Side Conditions,Discrete Mathematics 29 (1980), 135–159.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    G. Cornuéjols andW. Pulleyblank, Perfect Triangle-Free 2-matchings,Mathematical Programming Study 13 (1980), 1–7.MATHGoogle Scholar
  4. [4]
    G. Cornuéjols andW. Pulleyblank, The Travelling Salesman Polytope and {0,2}-matchings, inAnnals of Discrete Mathematics 16 (1982), 27–55.Google Scholar
  5. [5]
    J. Edmonds, Paths, Trees and Flowers,Canadian Journal of Mathematics 17 (1965), 449–467.MATHMathSciNetGoogle Scholar
  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.MathSciNetGoogle Scholar
  7. [7]
    L. Lovász, A Note on Factor-Critical Graphs,Studia Scientiarum Mathematicarum Hungarica 7 (1972), 279–280.MathSciNetGoogle Scholar
  8. [8]
    L. Lovász,Combinatorial Problems and Exercises, North Holland, 1979.Google Scholar
  9. [9]
    L. Lovász,Private Communication.Google Scholar
  10. [10]
    W. Pulleyblank,Faces of Matching Polyhedra, Ph. D. Thesis, University of Waterloo (1973).Google Scholar
  11. [11]
    W. Pulleyblank, Minimum Node Covers and 2-Bicritical Graphs,Mathematical Programming 17 (1979), 91–103.MATHCrossRefMathSciNetGoogle Scholar
  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.Google Scholar
  13. [13]
    W. T. Tutte, The Factorization of Linear Graphs,Journal of the London Mathematical Society 22 (1947), 107–111.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    W. T. Tutte, The Factors of Graphs,Canadian Journal of Mathematics 4 (1952), 314–328.MATHMathSciNetGoogle Scholar

Copyright information

© Akadémiai Kiadó 1983

Authors and Affiliations

  • G. Cornuéjols
    • 1
  • W. R. Pulleyblank
    • 2
  1. 1.G.S.I.A., Carnegie-Mellon UniversityPittsburghUSA
  2. 2.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

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