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 isr 2−r+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.
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References
A. Björner, Shellable and Cohen-Macaulay partially ordered sets,Trans. Am. Math. Soc.,260 (1980), 159–183.
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.
A. Björner, On complements in lattices of finite length,Discrete Mathematics,36 (1981), 325–326.
A. Björner, B. Korte andL. Lovász, Homotopy properties of greedoids,Advances in Applied Mathematics,6 (1985), 447–494.
H. Crapo, Selectors. A theory of formal languages, semimodular lattices, branchings and shelling processes,Advances in Math.,54 (1984), 233–277.
P. Edelman andR. Jamison, The theory of convex geometries,Geometriae Dedicata,19 (1985), 247–270.
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.
B. Korte andL. Lovász, Structural properties of greedoids,Combinatorica 3 (1983), 359–374.
B. Korte andL. Lovász, Greedoids and linear objective functions,SIAM J. Alg. Disc. Math. 5 (1984), 229–238.
B. Korte andL. Lovász, Basis graphs of greedoids and two-connectivity,Mathematical Programming Study 24 (1985), 158–165.
B. Korte andL. Lovász, Polymatroid greedoids,Journal of Combinatorial Theory, Series B 38 (1985), 41–72.
W.Schmidt,Strukturelle Aspekte in der kombinatorischen Optimierung: Greedoide auf Graphen, Dissertation, Bonn 1985.
A.Björner, G. M.Ziegler, Introduction to greedoids, to appear in: (N. White (ed.)),Combinatorial Geometries: Advanced Theory, Cambridge University.