A Generalization of Tutte's Characterization of Totally Unimodular Matrices

@article{Geelen1997AGO,
  title={A Generalization of Tutte's Characterization of Totally Unimodular Matrices},
  author={James F. Geelen},
  journal={J. Comb. Theory, Ser. B},
  year={1997},
  volume={70},
  pages={101-117}
}
  • J. Geelen
  • Published 1 May 1997
  • Mathematics
  • J. Comb. Theory, Ser. B
We characterize the symmetric (0, 1)-matrices that can be signed symmetrically so that every principal submatrix has determinant 0, ±1. This characterization generalizes Tutte's famous characterization of totally unimodular matrices. The result can be viewed as an excluded minor theorem for an interesting class of delta-matroids. 
View via Publisher
math.uwaterloo.ca
24 Citations
Highly Influential Citations
6
Background Citations
17
Methods Citations
3

Figures from this paper

24 Citations

A note on the spectra of certain skew-symmetric {1, 0, -1}-matrices
  • G. Wesp
  • Mathematics
    Discret. Math.
  • 2002
Binary matroids and local complementation
Principally Unimodular Skew-Symmetric Matrices
TLDR
This paper generalizes the well-known result of Camion, that if a (0, 1)-matrix can be signed to be totally unimodular then the signing is unique up to multiplying certain rows and columns by −1.
Nullity and Loop Complementation for Delta-Matroids
TLDR
It is shown that the symmetric-difference distance measure for set systems, and more specifically for delta-matroids, corresponds to the notion of nullity for symmetric and skew-symmetric matrices and is characterized in terms of equicardinality of minimal sets with respect to inclusion.
Isotropic Matroids I: Multimatroids and Neighborhoods
TLDR
Specific results of the latter type include a characterization of graphs that are locally equivalent to bipartite graphs, a direct proof that two forests are isomorphic if and only if their isotropic matroids are isomorph, and a way to express local equivalence indirectly, using only edge pivots.
Delta-matroids for graph theorists
TLDR
This survey provides an introduction to delta-matroids and aims to illustrate the two-way interaction between graph theory and deltamatroid theory that enriches both subjects.
Nullity Invariance for Pivot and the Interlace Polynomial
Integral Solutions of Linear Complementarity Problems
TLDR
It is shown that if M is rank-symmetric and principally unimodular, and q is integral, then the problem has an integral solution if it has a solution, and the results can be regarded as extensions of well-known results on integral solutions to linear programs.
Interlace Polynomials for Delta-Matroids
We show that the (single variable) interlace polynomial and several related graph polynomials may be defined for set systems (or more specifically, delta-matroids) in general. In this way, and using…
Combinatorial optimization : packing and covering
Preface 1. Clutters 2. T-Cuts and T-Joins 3. Perfect Graphs and Matrices 4. Ideal Matrices 5. Odd Cycles in Graphs 6. 0,+1 Matrices and Integral Polyhedra 7. Signing 0,1 Matrices to Be Totally…
...
...

References

SHOWING 1-10 OF 13 REFERENCES
A short proof of Tutte's characterization of totally unimodular matrices
Unimodularity and circle graphs
Greedy algorithm and symmetric matroids
TLDR
These set systems are characterized by a greedy algorithm and they are suitable for dealing with autodual properties of matroids and applications are given to the eulerian tours of 4-regular graphs and the theory ofg-matroids.
Alpha-balanced graphs and matrices and GF(3)-representability of matroids
A characterization of unimodular orientations of simple graphs
Representability of △-matroids over GF(2)
Lectures on matroids
Linear Complementarity Problem.
  • 1992
A homotopy theorem for matroids. II
Representability of 2-matroids
  • Colloq. Soc. Janos Bolyia
  • 1988
...
...