Skip to main content
SpringerLink
Log in

The Tutte decomposition

  • Chapter
A Source Book in Matroid Theory

  • 506 Accesses

Abstract

“Simple ideas are often the most powerful.” This adage is best exemplified in matroid theory by the method of Tutte (-Grothendieck) decomposition. This method has its origin in the following recursion formula (due to Foster1); see the concluding note in Whitney [32]) for the chromatic polynomial P(L; λ) of a graph L. Let L be a graph and A an edge in L linking two distinct vertices. Let L′ A be the graph obtained from L by deleting the edge A and L″ A the graph obtained from L by contracting A. Then

$$P(L;{\mkern 1mu} \lambda ){\mkern 1mu} = {\mkern 1mu} P({L'_A};{\mkern 1mu} \lambda ){\mkern 1mu} - {\mkern 1mu} P({L''_A};{\mkern 1mu} \lambda ).$$

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arrowsmith, D.K. and Jaeger, F.: On the enumeration of chain in regular chain groups, J. Combin. Theory Ser. B 32(1982), 75–89.

    MathSciNet  MATH  Google Scholar 

  2. Bass, H.: Algebraic K-theory, Benjamin, New York, 1968.

    MATH  Google Scholar 

  3. Beissinger, J.S.: On external activity and inversions in trees, J. Combin. Theory Ser. B 33(1982), 87–92.

    MathSciNet  MATH  Google Scholar 

  4. Bender, E., Viennot, G. and Williamson, S. G.: Global analysis of the delete-contract recursion for graphs and matroids, Linear and Multilinear Algebra 15(1984), 133–160.

    MathSciNet  MATH  Google Scholar 

  5. Berman, G.: The dichromate and orientations of a graph, Canad. J. Math. 29(1977), 947–956.

    MathSciNet  MATH  Google Scholar 

  6. Berman, G.: Decomposition of graph functions, J. Combin. Theory Ser. B 25(1978), 151–165.

    MathSciNet  Google Scholar 

  7. Brylawski, T.: A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154(1971), 1–22.

    MathSciNet  MATH  Google Scholar 

  8. Brylawski, T.: The Tutte-Grothendieck ring, Algebra Universalis 2(1972), 375–388.

    MathSciNet  MATH  Google Scholar 

  9. Brylawski, T.: A decomposition for combinatorial geometries, Trans. Amer. Math. Soc. 171(1972), 235–282.

    MathSciNet  MATH  Google Scholar 

  10. Brylawski, T.: Hyperplane reconstruction of the Tutte polynomial of a geometric lattice, Discrete Math. 35(1981), 25–38.

    MathSciNet  MATH  Google Scholar 

  11. Brylawski, T.: The Tutte polynomial, Part I: General theory, Matroid Theory and its Applications (Proc. C.I.M.E., Varenna, 1980), Liguori, Naples, 1982.

    Google Scholar 

  12. Crapo, H.H.: A higher invariant for matroids, J. Combin. Theory 2(1967), 406–417.

    MathSciNet  MATH  Google Scholar 

  13. Crapo, H.H.: The Tutte polynomial, Aequationes Math. 3(1969), 211–229.

    MathSciNet  MATH  Google Scholar 

  14. Farrell, E.J.: On a general class of graph polynomials, J. Combin. Theory Ser. B 26(1979), 111–122.

    MathSciNet  MATH  Google Scholar 

  15. Foster, R.M.: Geometrical circuits of electrical networks, Trans. Amer. Inst. Elect. Engrs. 52(1932), 309–317.

    Google Scholar 

  16. Foster, R.M.: Topologic and algebraic considerations in network synthesis, Proc. Sympos. Modern Network Synthesis, New York, 1952, pp. 8–18, Polytechnic Inst. of Brooklyn, New York, 1952.

    Google Scholar 

  17. Foster, R.M.: Academic and theoretical aspects of circuit theory, Proc. IRE 50(1962), 866–871.

    MathSciNet  Google Scholar 

  18. Heron, A.P.: Matroid polynomials, Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), pp. 164–202, Inst. Math. Appl., Southend-on-Sea, 1972.

    Google Scholar 

  19. Joyce, D.: Generalized chromatic polynomials, Discrete Math. 50(1984), 51–62

    MathSciNet  MATH  Google Scholar 

  20. Kung, J.P.S.: The Rédei function of a relation, J. Combin. Theory Ser. A 29(1980), 287–296.

    MathSciNet  MATH  Google Scholar 

  21. Las Vergnas, M.: On the Tutte polynomial of a morphism of matroids, Combinatorics 79, Part I, Ann. Discrete Math. 8(1980), 7–20.

    MATH  Google Scholar 

  22. Martin, P.: Remarkable valuation of the dichromatic polynomial of planar multigraphs, J. Combin. Theory Ser. B 24(1978), 318–324.

    MathSciNet  MATH  Google Scholar 

  23. Oxley, J.G.: On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(1982), 267–277.

    MathSciNet  MATH  Google Scholar 

  24. Oxley, J.G. and Welsh, D.J.A.: The Tutte polynomial and percolation, Graph Theory and Related Topics (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1977), pp. 329–339, Academic Press, New York and London, 1979.

    Google Scholar 

  25. Rosenstiehl, P. and Read, R.C.: On the principal edge tripartition of a graph, Ann. Discrete Math. 3(1978), 195–226.

    MathSciNet  MATH  Google Scholar 

  26. Smith, C.A.B.: Map colourings and linear mappings, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), pp. 259–283, Academic Press, London. 1969.

    Google Scholar 

  27. Smith, C.A.B.: On Tutte’s dichromatic polynomial, Advances in Graph Theory (B. Bollobás, ed.) Ann. Discrete Math. 3(1978), 247–257.

    Google Scholar 

  28. Stanley, R.P.: A Brylawski decomposition for finite ordered sets, Discrete Math. 4(1973), 77–82.

    MathSciNet  MATH  Google Scholar 

  29. Tutte, W.T.: A contribution to the theory of chromatic polynomials, Canad. J. Math. 6(1954), 80–91

    MathSciNet  MATH  Google Scholar 

  30. Tutte, W.T.: On dichromatic polynomials, J. Combin. Theory 2(1967), 301–302.

    MathSciNet  MATH  Google Scholar 

  31. Tutte, W.T.: Codichromatic graphs, J. Combin. Theory Ser. B 16(1974), 168–175.

    MathSciNet  MATH  Google Scholar 

  32. Tutte, W.T.: The dichromatic polynomial, Proc. Fifth British Combinatorial Conf. (Univ. Aberdeen, Aberdeen, 1975), pp. 605–635, Congressus Numerantium no. 15, Utilitas Math., Winnipeg, Man., 1976.

    Google Scholar 

  33. Tutte, W.T.: All the king’s horses. A guide to reconstruction, Graph Theory and Related Topics (Proc. Conf, Univ. Waterloo, Waterloo, Ont., 1977), pp. 15–33, Academic Press, 1979.

    Google Scholar 

  34. Tutte, W.T.: 1-factors and polynomials, Europ. J. Combin. 1(1980). 77–87.

    MathSciNet  MATH  Google Scholar 

  35. Whitney, H.: The coloring of graphs, Annals of Math. (2) 33(1933), 688–718.

    Google Scholar 

  36. Asano, T., Nishizeki, T., Oxley, J.G. and Saito, N.: A note on the critical problem for matroids, Europ. J. Combin. 5(1984), 93–97.

    MathSciNet  MATH  Google Scholar 

  37. Blake, I.F. and Mullin, R.C.: An introduction to algebraic and combinatorial coding theory, Academic Press, New York, 1976.

    MATH  Google Scholar 

  38. Brylawski, T.: Intersection theory for embeddings of matroids into uniform geometries, Stud. Appl. Math. 61(1979), 211–244.

    MathSciNet  MATH  Google Scholar 

  39. Brylawski, T.: Intersection theory for graphs, J. Combin. Theory Ser. B 30 (1981), 233–246.

    MathSciNet  MATH  Google Scholar 

  40. Brylawski, T., Lo Re, P.M., Mazzocca, F. and Olanda, D.: Alcune applicazioni della teorie dell’ intersezione alle geometrie di Galois, Ricerche di Matematica 29(1980), 65–84.

    MathSciNet  MATH  Google Scholar 

  41. Crapo, H.H. and Rota, G.-C.: On the foundations of combinatorial theory: Combinatorial geometries (Prelim, ed.), M.I.T. Press, Cambridge, Mass., 1970.

    MATH  Google Scholar 

  42. Dowling, T.A.: Codes, packings and the critical problem, Atti del Convegno di Geometria Combinatoria e sue applicazioni (Univ. Perugia, Perugia, 1970), pp. 209–224, Ist. Mat. Univ. Perugia, 1971.

    Google Scholar 

  43. Jaeger, F.: Flows and generalized coloring theorems in graphs, J. Combin. Theory Ser. B 26(1979), 205–216.

    MathSciNet  MATH  Google Scholar 

  44. Jaeger, F.: A constructive approach to the critical problem for matroids, Europ. J. Combin. 2(1981), 137–144.

    MathSciNet  MATH  Google Scholar 

  45. Kung, J.P.S.: The Rédei function of a relation, J. Combin. Theory Ser. A 29 (1980), 287–296.

    MathSciNet  MATH  Google Scholar 

  46. Kung, J.P.S.: Growth rates and critical exponents of classes of binary geometries, Trans. Amer. Math. Soc. 293(1986), 837–859.

    MathSciNet  MATH  Google Scholar 

  47. Kung, J.P.S., Murty, M.R. and Rota, G.-C.: On the Rédei zeta function, J. Number Theory 12(1980), 421–536.

    MathSciNet  MATH  Google Scholar 

  48. Lindström, B.: On the chromatic number of regular matroids, J. Combin. Theory Ser. B 24(1978), 367–369.

    MathSciNet  MATH  Google Scholar 

  49. Matthews, K.R.: An example from power residues of the critical problem of Crapo and Rota, J. Number Theory 9(1977), 203–208.

    MathSciNet  MATH  Google Scholar 

  50. Mullin, R.C. and Stanton, R.G.: A covering problem in binary spaces of finite dimension, Graph Theory and Related Topics (Proc. Conf. Univ. Waterloo, Waterloo, Ont., 1977), pp. 315–327, Academic Press, New York, 1979.

    Google Scholar 

  51. Oxley, J.G.: Colouring, packing and the critical problem, Quart. J. Math. Oxford Ser. (2) 29(1978), 11–22.

    MathSciNet  MATH  Google Scholar 

  52. Oxley, J.G.: Cocircuits coverings and packings for binary matroids, Math. Proc. Cambridge Philos. Soc. 83(1978), 347–351.

    MathSciNet  MATH  Google Scholar 

  53. Oxley, J.G.: On a covering problem of Mullin and Stanton for binary matroids, Aequationes Math. 19(1980), 104–112.

    MathSciNet  Google Scholar 

  54. Oxley, J.G.: On a matroid identity, Discrete Math. 44(1983), 55–60.

    MathSciNet  MATH  Google Scholar 

  55. Seymour, P.D.: Packing and covering with matroid circuits, J. Combin. Theory Ser. B 28(1980), 237–242.

    MathSciNet  MATH  Google Scholar 

  56. Seymour, P.D.: Nowhere-zero 6-flows, J. Combin. Theory Ser. B 30(1981), 130–135.

    MathSciNet  MATH  Google Scholar 

  57. Tutte, W.T.: A contribution to the theory of chromatic polynomials, Canad. J. Math. 6(1954), 80–91.

    MathSciNet  MATH  Google Scholar 

  58. Tutte, W.T.: On the algebraic theory of graph colorings, J. Combin. Theory 1(1966), 15–50.

    MathSciNet  MATH  Google Scholar 

  59. Tutte, W.T.: A geometrical version of the four color problem, Combinatorial Mathematics and its Applications (Proc. Conf., Univ. North Carolina, Chapel Hill, NC, 1967), pp. 553–560, Univ. North Carolina Press, Chapel Hill, N.C., 1969.

    Google Scholar 

  60. Tutte, W.T.: Projective geometry and the 4-color problem, Recent Progress in Combinatorics (Proc. Third Waterloo Conf. on Combinatorics, 1968), pp. 199–207, Academic Press, New York, 1969.

    Google Scholar 

  61. Walton, P.N. and Welsh, D.J.A.: On the chromatic number of binary matroids, Mathematika 27(1980), 1–9.

    MathSciNet  MATH  Google Scholar 

  62. Welsh, D.J.A.: Colouring problems and matroids, Surveys in Combinatorics (Proc. Seventh British Combinatorial Conf, Cambridge, 1979), pp. 229–257, London Math. Soc. Lecture Note Ser. No. 38, Cambridge Univ. Press, Cambridge, 1979.

    Google Scholar 

  63. Welsh, D.J.A.: Colourings, flows and projective geometry, Nieuw Arch. Wisk. (3) 28(1980), 159–176.

    MathSciNet  MATH  Google Scholar 

  64. Whittle, G.P.: On the critical exponents of transversal matroids, J. Combin. Theory Ser. B 37(1984), 94–95.

    MathSciNet  MATH  Google Scholar 

  65. Brylawski, T.: A combinatorial perspective on the Radon convexity theorem, Geometriae Dedicata 5(1976), 459–466.

    MathSciNet  MATH  Google Scholar 

  66. Cartier, P.: Les arrangements d’hyperplanes: Un chapitre de géométrie combinatoire, Séminaire Bourbaki, 1980/81, pp. 1–22 (No. 561), Lecture Notes in Math., Vol. 901, Springer-Verlag, Berlin and New York, 1981.

    Google Scholar 

  67. Cordovil, R.: Sur l’évaluation t(M; 2, 0) du polynôme de Tutte d’un matroïde et une conjecture de B. Grünbaum relative aux arrangement de droites du plan, Europ. J. Combin. 1(1980), 317–322.

    MathSciNet  MATH  Google Scholar 

  68. Cordovil, R., Las Vergnas, M. and Mandel, A.: Euler’s relations, Möbius functions and matroid identities, Geometriae Dedicata 12(1982), 147-162.

    Google Scholar 

  69. Edelman, P.: A partial order on the regions of Rn dissected by hyperplanes, Trans. Amer. Math. Soc., 283(1984), 617–632.

    MathSciNet  MATH  Google Scholar 

  70. Good, I.J. and Tideman, T.N.: Stirling numbers and a geometric structure from voting theory, J. Combin. Theory Ser. A 23(1977), 34–45.

    MathSciNet  MATH  Google Scholar 

  71. Greene, C. and Zaslavsky, T.: On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs, Trans. Amer. Math. Soc. 280(1983), 97–126.

    MathSciNet  MATH  Google Scholar 

  72. Jambu M. and Terao, H.: Free arrangements of hyperplanes and supersolvable lattices, Advances in Math. 52(1984), 248–258.

    MathSciNet  MATH  Google Scholar 

  73. Orlik, P. and Solomon, L.: Combinatorics and topology of complements of hyperplanes, Invent. Math. 56(1980), 167–189.

    MathSciNet  MATH  Google Scholar 

  74. Terao, H.: Arrangement of hyperplanes and their freeness, I and II, J. Faculty Sci., Univ. Tokyo, Sci. IA, 27(1980), 293–312 and 313–320.

    MathSciNet  MATH  Google Scholar 

  75. Terao, H.: Generalized exponents of a free arrangements of hyperplanes and Shepard-Todd-Brieskorn formula, Invent. Math. 63(1981), 159–179.

    MathSciNet  MATH  Google Scholar 

  76. Zaslavsky, T.: Maximal dissections of a simplex, J. Combin. Theory Ser. A 20(1976), 244–257.

    MathSciNet  MATH  Google Scholar 

  77. Zaslavsky, T.: A combinatorial analysis of topological dissections, Advances in Math. 25(1977), 267–285.

    MathSciNet  MATH  Google Scholar 

  78. Zaslavsky, T.: Arrangements of hyperplanes; matroids and graphs, Proc. Tenth. S.E. Conf. on Combinatorics, Graph Theory and Computing (Boca Raton, 1979), Vol. II, pp. 895–911, Utilitas Math., Winnipeg, Man., 1979.

    Google Scholar 

  79. Zaslavsky, T.: The geometry of root systems and signed graphs, Amer. Math. Monthly 88(1981), 88–105.

    MathSciNet  MATH  Google Scholar 

  80. Zaslavsky, T.: The slimmest arrangements of hyperplanes: I. Geometric lattices and projective arrangements, Geometriae Dedicata 14(1983), 243–259.

    MathSciNet  MATH  Google Scholar 

  81. Zaslavsky, T.: The slimmest arrangements of hyperplanes: II. Basepointed geometric lattices and Euclidean arrangements, Mathematika 28(1981), 169–190

    MathSciNet  MATH  Google Scholar 

  82. E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York, 1968.

    MATH  Google Scholar 

  83. T. Brylawski, A decomposition for combinatorial geometries, to be published.

    Google Scholar 

  84. H. Crapo, Möbius inversion in lattices, Arch. Math. 19, 38 (1968).

    MathSciNet  Google Scholar 

  85. H. Crapo, The Tutte polynomial, Acquationes Math. 3, 211 (1969).

    MathSciNet  MATH  Google Scholar 

  86. H. Crapo and G.-C. Rota, Combinatorial Geometries, M.I.T. Press, Cambridge, 1971.

    Google Scholar 

  87. T. Dowling, Codes, packing, and the critical problem, Atti del Convegno di Geometria Combinatoria e sua Applicazioni, Perugia, 1971, p. 209.

    Google Scholar 

  88. J. H. Van Lint, Coding Theory, Springer, Heidelberg, 1971.

    MATH  Google Scholar 

  89. J. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell Syst. Tech. J. 42, 654 (1962).

    MathSciNet  Google Scholar 

  90. W. T. Tutte, A ring in graph theory, Proc. Camb. Phil. Soc. 43, 26 (1947).

    MathSciNet  MATH  Google Scholar 

  91. W. T. Tutte, On dichromatic polynomials, J. Comb. Theory 2, 301 (1967).

    MathSciNet  MATH  Google Scholar 

  92. W. T. Tutte, On the algebraic theory of graph coloring, J. Comb. Theory 1, 15 (1966).

    MathSciNet  MATH  Google Scholar 

  93. O. Veblen, An application of modular equations in analysis situs, Ann. Math. 14, 86 (1912).

    MathSciNet  MATH  Google Scholar 

  94. H. Whitney, On the abstract properties of linear dependence, Am. J. Math. 57, 509 (1935).

    MathSciNet  Google Scholar 

  95. Birkhoff, G.: Lattice Theory, Third Edition, Amer. Math. Soc. Colloq. Publ., Vol. 25, Amer. Math. Soc., Providence, R.I. 1967. MR 37 #2638.

    Google Scholar 

  96. Brylawski, T.: A decomposition for combinatorial geometries, Trans. Amer. Math. Soc. 171(1972), 235–282. MR 46 #8869.

    MathSciNet  MATH  Google Scholar 

  97. Buck, R.C.: Partition of space, Amer. Math. Monthly 50(1943), 541–544. MR 5, 105.

    MathSciNet  MATH  Google Scholar 

  98. Crapo, H.H.: Möbius inversion in lattices, Arch. Math. (Basel) 19(1968), 595–607. MR 39 #6791.

    MathSciNet  Google Scholar 

  99. Crapo, H.H. and Rota, G.-C.: On the Foundations of Combinatorial Theory: Combinatorial Geometries (Prelim. Ed.), M.I.T. Press, Cambridge, Mass., 1970. MR 45 #74.

    MATH  Google Scholar 

  100. Grünbaum, B.: Convex Polytopes, Interscience, New York, 1967. MR 37 #2085. (See Chapter 18.)

    MATH  Google Scholar 

  101. Grünbaum, B.: Arrangements of hyperplanes, Proc. Second Louisiana Conf. on Combinatorics, Graph Theory, and Computing (R.C. Mullin et al., eds.), Baton Rouge, 1971.

    Google Scholar 

  102. Grünbaum, B.: Arrangements and Spreads, CBMS Regional Conference Series in Mathematics, No. 10, Amer. Math. Soc., Providence, R.I., 1972. MR46 #6148.

    Google Scholar 

  103. Harding, E.F.: The number of partitions of a set of N points in k dimensions induced by hyperplanes, Proc. Edinburgh Math. Soc. (2)15(1966/67), 285–289. MR 37 #4702.

    MathSciNet  Google Scholar 

  104. Roberts, S.: On the figures formed by the intercepts of a system of straight lines in a plane, and on analogous relations in space of three dimensions, Proc. London Math. Soc. 19(1888), 405–422.

    MATH  Google Scholar 

  105. Rota, G.-C.: On the foundations of combinatorial theory, I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2(1964), 340–368. MR 30 #4688.

    MathSciNet  MATH  Google Scholar 

  106. Schläfli, L.: Theorie der vielfachen Kontinuität, reprinted in his Gesammelte mathematische Abhandlungen, Band I, Birkhäuser, Basel, 1950. (Written in 1850–52.)

    Google Scholar 

  107. Smith, J.: Arranging Hyperplanes in the Home, Pan, London, 1926.

    Google Scholar 

  108. Steiner, J.: Einige Gesetze über die Theilung der Ebene und des Raumes, J. Reine Angew. Math. 1(1826), 349–364.

    MATH  Google Scholar 

  109. Winder, R.O.: Partitions of N-space by hyperplanes, SIAM J. Appl. Math. 14(1966), 811–818. MR 34 #8281.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Trinity College, Cambridge, USA

    W. T. Tutte

  2. Massachusetts Institute of Technology, USA

    Curtis Greene

Authors
  1. W. T. Tutte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Curtis Greene
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas Zaslavsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer Science+Business Media New York

About this chapter

Cite this chapter

Tutte, W.T., Greene, C., Zaslavsky, T. (1986). The Tutte decomposition. In: A Source Book in Matroid Theory. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4684-9199-9_4

Download citation

  • DOI: https://doi.org/10.1007/978-1-4684-9199-9_4

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-0-8176-3173-4

  • Online ISBN: 978-1-4684-9199-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation