Brick decompositions and the matching rank of graphs


The number of linearly independent perfect matchings of a graph — or, equivalently, the dimension of the perfect matching polytope — is determined in various senses. First it is shown that the fact that every linear objective function can be optimized over the perfect matchings in polynomial time implies the existence of a polynomial-time algorithm to determine the dimension of this set. This observation also yields polynomial algorithms to determine, among others, the number of linearly independent common bases of two matroids and the number of linearly independent maximum stable sets in claw-free or perfect graphs. For the case of perfect matchings, Naddef’s minimax theorem for the dimension of the perfect matching polytope is strengthened and it is shown how the decomposition theory of matchings in graphs can be applied to derive a particularly simple formula for this dimension. This formula is based upon the number of constituents of a certain decomposition of the graph which we call a brick decomposition. Finally, these results are applied to obtain a description of the facets of the perfect matching polytope.

This is a preview of subscription content, access via your institution.


  1. [1]

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

    MathSciNet  Google Scholar 

  2. [2]

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

    MATH  MathSciNet  Google Scholar 

  3. [3]

    J. Edmonds, Matroid intersection,Annals of Discrete Mathematics 4 (1979) 39–49.

    MATH  MathSciNet  Article  Google Scholar 

  4. [4]

    J. Edmonds, Systems of distinct representatives and linear algebra,Journal of Research of the National Bureau of Standards 71B (1967) 241–245.

    MathSciNet  Google Scholar 

  5. [5]

    T. Gallai, Maximale Systeme unabhängiger Kanten,Mat. Kut. Int. Közl. 9 (1964) 353–395.

    Google Scholar 

  6. [6]

    M. Grötschel, L. Lovász andA. Schrijver, The ellipsoid method and its consequences for combinatorial optimization,Combinatorica 1(2) (1981) 169–197.

    MATH  MathSciNet  Google Scholar 

  7. [7]

    A. Kotzig, Ein Beitrag zur Theorie der endlichen Graphen mit linearen Faktoren I—II—III (in Slovak with a Germany summary),Math. Fyz. Casopis,9 (1959) pp. 83–91, 136–159,10 (1960) 205–215.

    Google Scholar 

  8. [8]

    L. Lovász, On the structure of factorizable graphs,Acta Math. Acad. Sci. Hung. 23 (1972) 179–195.

    MATH  Article  Google Scholar 

  9. [9]

    L. Lovász andM. D. Plummer, On bicritical graphs,Infinite and Finite Sets, Colloqu. Math. Soc. J. Bolyai10, Budapest, (A. Hajnal, R. Rado and V. T. Sós et al eds) (1975) 1051–1079.

  10. [10]

    G. Minty, On maximal independent sets of vertices in claw-free graphs,Journal of Combinatorial Theory, Series B 28 (1980) 284–304.

    MATH  Article  MathSciNet  Google Scholar 

  11. [11]

    D. Naddef, Rank of maximum matchings of a graph,Mathematical Programming 22 (1982) 52–70.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

    D. Naddef andW. R. Pulleyblank, Matchings in regular graphs,Discrete Mathematics 34 (1981) 283–290.

    MATH  Article  MathSciNet  Google Scholar 

  13. [13]

    D. Naddef andW. R. Pulleyblank, On GF2 rank and ear decomposition of elementary graphs,Annals of Discrete Mathematics 16 (1982) 285–304.

    MathSciNet  Google Scholar 

  14. [14]

    W. R. Pulleyblank, The matching rank of Halin graphs,Report No. 80165-O R, Inst. für Operations Research, Universität Bonn (1980).

  15. [15]

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

Download references

Author information



Additional information

Dedicated to Tibor Gallai on his seventieth birthday

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Edmonds, J., Pulleyblank, W.R. & Lovász, L. Brick decompositions and the matching rank of graphs. Combinatorica 2, 247–274 (1982).

Download citation

AMS subject classification (1980)

  • 05 C 35
  • 90 C 05
  • 05 B 35