The Tutte decomposition

  • W. T. Tutte
  • Curtis Greene
  • Thomas Zaslavsky


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


Isomorphism Class Linear Code Critical Problem Simple Closed Curve Linear Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.MathSciNetMATHGoogle Scholar
  2. Bass, H.: Algebraic K-theory, Benjamin, New York, 1968.MATHGoogle Scholar
  3. Beissinger, J.S.: On external activity and inversions in trees, J. Combin. Theory Ser. B 33(1982), 87–92.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar
  5. Berman, G.: The dichromate and orientations of a graph, Canad. J. Math. 29(1977), 947–956.MathSciNetMATHGoogle Scholar
  6. Berman, G.: Decomposition of graph functions, J. Combin. Theory Ser. B 25(1978), 151–165.MathSciNetGoogle Scholar
  7. Brylawski, T.: A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154(1971), 1–22.MathSciNetMATHGoogle Scholar
  8. Brylawski, T.: The Tutte-Grothendieck ring, Algebra Universalis 2(1972), 375–388.MathSciNetMATHGoogle Scholar
  9. Brylawski, T.: A decomposition for combinatorial geometries, Trans. Amer. Math. Soc. 171(1972), 235–282.MathSciNetMATHGoogle Scholar
  10. Brylawski, T.: Hyperplane reconstruction of the Tutte polynomial of a geometric lattice, Discrete Math. 35(1981), 25–38.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar
  13. Crapo, H.H.: The Tutte polynomial, Aequationes Math. 3(1969), 211–229.MathSciNetMATHGoogle Scholar
  14. Farrell, E.J.: On a general class of graph polynomials, J. Combin. Theory Ser. B 26(1979), 111–122.MathSciNetMATHGoogle 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.MathSciNetGoogle 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), 51–62MathSciNetMATHGoogle Scholar
  20. Kung, J.P.S.: The Rédei function of a relation, J. Combin. Theory Ser. A 29(1980), 287–296.MathSciNetMATHGoogle 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.MATHGoogle Scholar
  22. Martin, P.: Remarkable valuation of the dichromatic polynomial of planar multigraphs, J. Combin. Theory Ser. B 24(1978), 318–324.MathSciNetMATHGoogle Scholar
  23. Oxley, J.G.: On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(1982), 267–277.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar
  29. Tutte, W.T.: A contribution to the theory of chromatic polynomials, Canad. J. Math. 6(1954), 80–91MathSciNetMATHGoogle Scholar
  30. Tutte, W.T.: On dichromatic polynomials, J. Combin. Theory 2(1967), 301–302.MathSciNetMATHGoogle Scholar
  31. Tutte, W.T.: Codichromatic graphs, J. Combin. Theory Ser. B 16(1974), 168–175.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar
  37. Blake, I.F. and Mullin, R.C.: An introduction to algebraic and combinatorial coding theory, Academic Press, New York, 1976.MATHGoogle Scholar
  38. Brylawski, T.: Intersection theory for embeddings of matroids into uniform geometries, Stud. Appl. Math. 61(1979), 211–244.MathSciNetMATHGoogle Scholar
  39. Brylawski, T.: Intersection theory for graphs, J. Combin. Theory Ser. B 30 (1981), 233–246.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle 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.MATHGoogle 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.MathSciNetMATHGoogle Scholar
  44. Jaeger, F.: A constructive approach to the critical problem for matroids, Europ. J. Combin. 2(1981), 137–144.MathSciNetMATHGoogle Scholar
  45. Kung, J.P.S.: The Rédei function of a relation, J. Combin. Theory Ser. A 29 (1980), 287–296.MathSciNetMATHGoogle Scholar
  46. Kung, J.P.S.: Growth rates and critical exponents of classes of binary geometries, Trans. Amer. Math. Soc. 293(1986), 837–859.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar
  48. Lindström, B.: On the chromatic number of regular matroids, J. Combin. Theory Ser. B 24(1978), 367–369.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar
  52. Oxley, J.G.: Cocircuits coverings and packings for binary matroids, Math. Proc. Cambridge Philos. Soc. 83(1978), 347–351.MathSciNetMATHGoogle Scholar
  53. Oxley, J.G.: On a covering problem of Mullin and Stanton for binary matroids, Aequationes Math. 19(1980), 104–112.MathSciNetGoogle Scholar
  54. Oxley, J.G.: On a matroid identity, Discrete Math. 44(1983), 55–60.MathSciNetMATHGoogle Scholar
  55. Seymour, P.D.: Packing and covering with matroid circuits, J. Combin. Theory Ser. B 28(1980), 237–242.MathSciNetMATHGoogle Scholar
  56. Seymour, P.D.: Nowhere-zero 6-flows, J. Combin. Theory Ser. B 30(1981), 130–135.MathSciNetMATHGoogle Scholar
  57. Tutte, W.T.: A contribution to the theory of chromatic polynomials, Canad. J. Math. 6(1954), 80–91.MathSciNetMATHGoogle Scholar
  58. Tutte, W.T.: On the algebraic theory of graph colorings, J. Combin. Theory 1(1966), 15–50.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar
  64. Whittle, G.P.: On the critical exponents of transversal matroids, J. Combin. Theory Ser. B 37(1984), 94–95.MathSciNetMATHGoogle Scholar
  65. Brylawski, T.: A combinatorial perspective on the Radon convexity theorem, Geometriae Dedicata 5(1976), 459–466.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar
  72. Jambu M. and Terao, H.: Free arrangements of hyperplanes and supersolvable lattices, Advances in Math. 52(1984), 248–258.MathSciNetMATHGoogle Scholar
  73. Orlik, P. and Solomon, L.: Combinatorics and topology of complements of hyperplanes, Invent. Math. 56(1980), 167–189.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar
  75. Terao, H.: Generalized exponents of a free arrangements of hyperplanes and Shepard-Todd-Brieskorn formula, Invent. Math. 63(1981), 159–179.MathSciNetMATHGoogle Scholar
  76. Zaslavsky, T.: Maximal dissections of a simplex, J. Combin. Theory Ser. A 20(1976), 244–257.MathSciNetMATHGoogle Scholar
  77. Zaslavsky, T.: A combinatorial analysis of topological dissections, Advances in Math. 25(1977), 267–285.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar
  80. Zaslavsky, T.: The slimmest arrangements of hyperplanes: I. Geometric lattices and projective arrangements, Geometriae Dedicata 14(1983), 243–259.MathSciNetMATHGoogle Scholar
  81. Zaslavsky, T.: The slimmest arrangements of hyperplanes: II. Basepointed geometric lattices and Euclidean arrangements, Mathematika 28(1981), 169–190MathSciNetMATHGoogle Scholar
  82. 1.
    E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York, 1968.MATHGoogle Scholar
  83. 2.
    T. Brylawski, A decomposition for combinatorial geometries, to be published.Google Scholar
  84. 3.
    H. Crapo, Möbius inversion in lattices, Arch. Math. 19, 38 (1968).MathSciNetGoogle Scholar
  85. 4.
    H. Crapo, The Tutte polynomial, Acquationes Math. 3, 211 (1969).MathSciNetMATHGoogle Scholar
  86. 5.
    H. Crapo and G.-C. Rota, Combinatorial Geometries, M.I.T. Press, Cambridge, 1971.Google Scholar
  87. 6.
    T. Dowling, Codes, packing, and the critical problem, Atti del Convegno di Geometria Combinatoria e sua Applicazioni, Perugia, 1971, p. 209.Google Scholar
  88. 7.
    J. H. Van Lint, Coding Theory, Springer, Heidelberg, 1971.MATHGoogle Scholar
  89. 8.
    J. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell Syst. Tech. J. 42, 654 (1962).MathSciNetGoogle Scholar
  90. 9.
    W. T. Tutte, A ring in graph theory, Proc. Camb. Phil. Soc. 43, 26 (1947).MathSciNetMATHGoogle Scholar
  91. 10.
    W. T. Tutte, On dichromatic polynomials, J. Comb. Theory 2, 301 (1967).MathSciNetMATHGoogle Scholar
  92. 11.
    W. T. Tutte, On the algebraic theory of graph coloring, J. Comb. Theory 1, 15 (1966).MathSciNetMATHGoogle Scholar
  93. 12.
    O. Veblen, An application of modular equations in analysis situs, Ann. Math. 14, 86 (1912).MathSciNetMATHGoogle Scholar
  94. 13.
    H. Whitney, On the abstract properties of linear dependence, Am. J. Math. 57, 509 (1935).MathSciNetGoogle Scholar
  95. [LT3]
    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. [DCG]
    Brylawski, T.: A decomposition for combinatorial geometries, Trans. Amer. Math. Soc. 171(1972), 235–282. MR 46 #8869.MathSciNetMATHGoogle Scholar
  97. [P]
    Buck, R.C.: Partition of space, Amer. Math. Monthly 50(1943), 541–544. MR 5, 105.MathSciNetMATHGoogle Scholar
  98. [MI]
    Crapo, H.H.: Möbius inversion in lattices, Arch. Math. (Basel) 19(1968), 595–607. MR 39 #6791.MathSciNetGoogle Scholar
  99. [CG]
    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.MATHGoogle Scholar
  100. [CP]
    Grünbaum, B.: Convex Polytopes, Interscience, New York, 1967. MR 37 #2085. (See Chapter 18.)MATHGoogle Scholar
  101. [AH]
    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. [AS]
    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. [N]
    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.MathSciNetGoogle Scholar
  104. [F]
    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.MATHGoogle Scholar
  105. [FCT]
    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.MathSciNetMATHGoogle Scholar
  106. [VK]
    Schläfli, L.: Theorie der vielfachen Kontinuität, reprinted in his Gesammelte mathematische Abhandlungen, Band I, Birkhäuser, Basel, 1950. (Written in 1850–52.)Google Scholar
  107. [AHH]
    Smith, J.: Arranging Hyperplanes in the Home, Pan, London, 1926.Google Scholar
  108. [T]
    Steiner, J.: Einige Gesetze über die Theilung der Ebene und des Raumes, J. Reine Angew. Math. 1(1826), 349–364.MATHGoogle Scholar
  109. [PH]
    Winder, R.O.: Partitions of N-space by hyperplanes, SIAM J. Appl. Math. 14(1966), 811–818. MR 34 #8281.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • W. T. Tutte
    • 1
  • Curtis Greene
    • 2
  • Thomas Zaslavsky
  1. 1.Trinity CollegeCambridgeUSA
  2. 2.Massachusetts Institute of TechnologyUSA

Personalised recommendations