Advertisement

Combinatorica

, Volume 8, Issue 2, pp 217–234 | Cite as

Branchings in rooted graphs and the diameter of greedoids

  • G. M. Ziegler
Article

Abstract

LetG be a 2-connected rooted graph of rankr andA, B two (rooted) spanning trees ofG We show that the maximum number of exchanges of leaves that can be required to transformA intoB isr2r+1 (r>0). This answers a question by L. Lovász.

There is a natural reformulation of this problem in the theory ofgreedoids, which asks for the maximum diameter of the basis graph of a 2-connected branching greedcid of rankr.

Greedoids are finite accessible set systems satisfying the matroid exchange axiom. Their theory provides both language and conceptual framework for the proof. However, it is shown that for general 2-connected greedoids (not necessarily constructed from branchings in rooted graphs) the maximum diameter is 2r−1.

AMS subject classification(1980)

05 B 35 05 c 20 68 Q 20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Björner, Shellable and Cohen-Macaulay partially ordered sets,Trans. Am. Math. Soc.,260 (1980), 159–183.Google Scholar
  2. [2]
    A. Björner, On matroids, groups and exchange languages, in: (L. Lovász and A. Recski (eds.)),Matroid theory and its applications. Conference Proceedings, Szeged, September 1982. Colloquia Mathematica Societatis János Bolyai40. North Holland, Amsterdam, Budapest, 1985.Google Scholar
  3. [3]
    A. Björner, On complements in lattices of finite length,Discrete Mathematics,36 (1981), 325–326.Google Scholar
  4. [4]
    A. Björner, B. Korte andL. Lovász, Homotopy properties of greedoids,Advances in Applied Mathematics,6 (1985), 447–494.Google Scholar
  5. [5]
    H. Crapo, Selectors. A theory of formal languages, semimodular lattices, branchings and shelling processes,Advances in Math.,54 (1984), 233–277.Google Scholar
  6. [6]
    P. Edelman andR. Jamison, The theory of convex geometries,Geometriae Dedicata,19 (1985), 247–270.Google Scholar
  7. [7]
    B. Korte andL. Lovász, Greedoids, a structural framework for the greedy algorithm, in: (W. R. Pulleyblank (ed.)),Progress in Combinatorial Optimization. Proceedings of the Silver Jubilee Conference on Combinatorics, Waterloo, June 1982. Academic Press, London-New York-San Francisco 1984, 221–243.Google Scholar
  8. [8]
    B. Korte andL. Lovász, Structural properties of greedoids,Combinatorica 3 (1983), 359–374.Google Scholar
  9. [9]
    B. Korte andL. Lovász, Greedoids and linear objective functions,SIAM J. Alg. Disc. Math. 5 (1984), 229–238.Google Scholar
  10. [10]
    B. Korte andL. Lovász, Basis graphs of greedoids and two-connectivity,Mathematical Programming Study 24 (1985), 158–165.Google Scholar
  11. [11]
    B. Korte andL. Lovász, Polymatroid greedoids,Journal of Combinatorial Theory, Series B 38 (1985), 41–72.Google Scholar
  12. [12]
    W.Schmidt,Strukturelle Aspekte in der kombinatorischen Optimierung: Greedoide auf Graphen, Dissertation, Bonn 1985.Google Scholar
  13. [13]
    A.Björner, G. M.Ziegler, Introduction to greedoids, to appear in: (N. White (ed.)),Combinatorial Geometries: Advanced Theory, Cambridge University.Google Scholar

Copyright information

© Akadémiai Kiadó 1988

Authors and Affiliations

  • G. M. Ziegler
    • 1
  1. 1.M. I. T. 2-251CambridgeUSA

Personalised recommendations