Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T12:17:58.321Z Has data issue: false hasContentIssue false
  • English
  • Français

Multi-Path Matroids

Published online by Cambridge University Press:  01 March 2007

JOSEPH E. BONIN  and
OMER GIMÉNEZ
JOSEPH E. BONIN
Affiliation:
Department of Mathematics, The George Washington University, Washington, DC 20052, USA (e-mail: jbonin@gwu.edu)
OMER GIMÉNEZ
Affiliation:
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Jordi Girona 1–3, 08034, Barcelona, Spaine (e-mail: omer.gimenez@upc.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce the minor-closed, dual-closed class of multi-path matroids. We give a polynomial-time algorithm for computing the Tutte polynomial of a multi-path matroid, we describe their basis activities, and we prove some basic structural properties. Key elements of this work are two complementary perspectives we develop for these matroids: on the one hand, multi-path matroids are transversal matroids that have special types of presentations; on the other hand, the bases of multi-path matroids can be viewed as sets of lattice paths in certain planar diagrams.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 2 , March 2007 , pp. 193 - 217
Copyright
Copyright © Cambridge University Press 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Björner, A. (1992) The homology and shellability of matroids and geometric lattices. In Matroid Applications (White, N., ed.), Cambridge University Press, pp. 226–283.CrossRefGoogle Scholar
[2]Bonin, J., de Mier, A. and Noy, M. (2003) Lattice path matroids: Enumerative aspects and Tutte polynomials. J. Combin. Theory Ser. A 104 63–94.CrossRefGoogle Scholar
[3]Bonin, J. and de Mier, A. Lattice path matroids: Structural aspects. European J. Combin., to appear.Google Scholar
[4]Brylawski, T. and Oxley, J. G. (1992) The Tutte polynomial and its applications. In Matroid Applications (White, N., ed.), Cambridge University Press, pp. 123–225.CrossRefGoogle Scholar
[5]Colbourn, C. J., Provan, J. S. and Vertigan, D. (1995) The complexity of computing the Tutte polynomial on transversal matroids. Combinatorica 15 1–10.CrossRefGoogle Scholar
[6]Edmonds, J. and Fulkerson, D. R. (1965) Transversals and matroid partition. J. Res. Nat. Bur. Standards Sect. B 69B 147–153.CrossRefGoogle Scholar
[7]Giménez, O. and Noy, M. (2006) On the complexity of computing the Tutte polynomial of bicircular matroids. Combin. Probab. Comput. 15 385–395.CrossRefGoogle Scholar
[8]Jaeger, F., Vertigan, D. and Welsh, D. J. A. (1990) On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Cambridge Philos. Soc. 108 35–53.CrossRefGoogle Scholar
[9]Oxley,, J. G. (1992) Matroid Theory, Oxford University Press.Google Scholar
[10]Vertigan, D. (1998) Bicycle dimension and special points of the Tutte polynomial. J. Combin. Theory Ser. B 74 378–396.CrossRefGoogle Scholar
[11]Vertigan, D. and Welsh, D. J. A. (1992) The computational complexity of the Tutte plane: The bipartite case. Combin. Probab. Comput. 1 181–187.CrossRefGoogle Scholar
[12]Welsh, D. J. A. (1993) Complexity: Knots, Colourings and Counting, Cambridge University Press.CrossRefGoogle Scholar