Partition coefficients of acyclic graphs

  • John L. Pfaltz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1017)


We develop the concept of a “closure space”; which appears with different names in many aspects of graph theory. We show that acyclic graphs can be almost characterized by the partition coefficients of their associated closure spaces. The resulting nearly total ordering of all acyclic graphs (or partial orders) provides an effective isomorphism filter and the basis for efficient retrieval in secondary storage. Closure spaces and their partition coefficients provide the theoretical basis for a new computer system being developed to investigate the properties of arbitrary acyclic graphs and partial orders.


Partition Coefficient Partial Order Closure Operator Transitive Closure Closure Space 
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  1. 1.
    G. D. Birkhoff and D. Lewis. Chromatic polynomials. Trans. Amer. Math. Soc., 60:355–451, 1946.Google Scholar
  2. 2.
    Richard A. Brualdi, Hyung Chan Jung, and William T. Trotter, Jr. On the poset of all posets on n elements. Discrete Applied Mathematics, 1994. To appear.Google Scholar
  3. 3.
    R.F. Churchhouse. Congruence properties of the binary partition function. Proc. Cambridge Phil. Soc., 66(2):371–376, 1969.Google Scholar
  4. 4.
    R.F. Churchhouse. Binary partitions. In A.O.L. Atkin and B.J. Birch, editors, Computers in Number Theory, pages 397–400. Academic Press, 1971.Google Scholar
  5. 5.
    Joseph C. Culberson and Gregory J. E. Rawlins. New results from an algorithm for counting posets. Order, 7:361–374, 1991.Google Scholar
  6. 6.
    Dragos Cvetkovic, Peter Rowlinson, and Slobodan Simic. A study of eigenspaces of graphs. Linear Algebra and Its Applic., 182:45–66, Mar. 1993.Google Scholar
  7. 7.
    Brenda L. Dietrich. Matroids and antimatroids — a survey. Discrete Mathematics, 78:223–237, 1989.Google Scholar
  8. 8.
    Feodor F. Dragan, Falk Nicolai, and Andreas Brandstadt. Convexisty and handfree graphs. Technical Report SM-DU-290, Gerhard-Mercator Univ., Duisburg, Germany, May 1995.Google Scholar
  9. 9.
    Paul H. Edelman. Meet-distributive lattices and the anti-exchange closure. Algebra Universalis, 10(3):290–299, 1980.Google Scholar
  10. 10.
    Paul H. Edelman and Robert E. Jamison. The theory of convex geometries. Geometriae Dedicata, 19(3):247–270, Dec. 1985.Google Scholar
  11. 11.
    Martin Farber and Robert E. Jamison. Convexity in graphs and hypergraphs. SIAM J. Algebra and Discrete Methods, 7(3):433–444, July 1986.Google Scholar
  12. 12.
    George Gratzer. General Lattice Theory. Academic Press, 1978.Google Scholar
  13. 13.
    Frank Harary. Graph Theory. Addison-Wesley, 1969.Google Scholar
  14. 14.
    Robert E. Jamison-Waldner. Partition numbers for trees and ordered sets. Pacific J. of Math., 96(1):115–140, Sept. 1981.Google Scholar
  15. 15.
    Bernhard Korte, Laszlo Lovasz, and Rainer Schrader. Greedoids. Springer-Verlag, Berlin, 1991.Google Scholar
  16. 16.
    K. Mahler. On a special functional equation. J. London Math. Soc., 15(58):115–123, Apr. 1940.Google Scholar
  17. 17.
    John L. Pfaltz. Convexity in directed graphs. J. of Comb. Theory, 10(2):143–162, Apr. 1971.Google Scholar
  18. 18.
    John L. Pfaltz. Computer Data Structures. McGraw-Hill, Feb. 1977.Google Scholar
  19. 19.
    John L. Pfaltz. Partitions of 2n. Technical Report TR CS-94-22, University of Virginia, June 1994.Google Scholar
  20. 20.
    John L. Pfaltz. Closure lattices. Discrete Mathematics, 1995. (to appear), preprint available as Tech. Rpt. CS-94-02 through home page Scholar
  21. 21.
    John L. Pfaltz. Partially ordering the subsets of a closure space. ORDER, 1995. (submitted).Google Scholar
  22. 22.
    A. J. Schwenk. Computing the characteristic polynomial of a graph. In R. Bari and F. Harary, editors, Graphs and Combinatorics, pages 153–172. Springer Verlag, 1974.Google Scholar
  23. 23.
    N. J. A. Sloane. A Handbook of Integer Sequences. Academic Press, 1973. On-line version at ''.Google Scholar
  24. 24.
    Richard P. Stanley. Enumerative Combinatorics, Vol 1. Wadsworth & Brooks/Cole, 1986.Google Scholar
  25. 25.
    W. T. Tutte. Introduction to the Theory of Matroids. Amer. Elsevier, 1971.Google Scholar
  26. 26.
    D.J.A. Welsh. Matroid Theory. Academic Press, 1976.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • John L. Pfaltz
    • 1
  1. 1.Dept. of Computer ScienceUniversity of VirginiaCharlottesville

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