# Enumeration of 2-Polymatroids on up to Seven Elements

@article{Savitsky2014EnumerationO2, title={Enumeration of 2-Polymatroids on up to Seven Elements}, author={Thomas J. Savitsky}, journal={SIAM J. Discret. Math.}, year={2014}, volume={28}, pages={1641-1650} }

A theory of single-element extensions of integer polymatroids analogous to that of matroids is developed. We present an algorithm to generate a catalog of 2-polymatroids, up to isomorphism. When we implemented this algorithm on a computer, obtaining all 2-polymatroids on at most seven elements, we discovered the surprising fact that the number of 2-polymatroids on seven elements fails to be unimodal in rank.

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- 2012

(1,−1) [dLL09]. (1, 2) [BKM08, DK06, DK10]. (1,≤ l) [Lai05]. (1,m+ 1, n+ 1) [BD01]. (1− ε) ex(n,C4) [BS10a]. (2 + )n [DZ01]. (3, 1)∗ [Xu09]. (7, 2) [CW09]. (9v, 4, 1) [FHMY01]. (d, k) [KS03a]. (k, l)…

## References

SHOWING 1-10 OF 28 REFERENCES

### Duality in Polymatroids and Set Functions

- MathematicsCombinatorics, Probability and Computing
- 1992

There is a unique involutary operation on the class of k-polymatroids that interchanges deletion and contraction, which generalise those of Kung [3].

### Single-element extensions of matroids

- Mathematics
- 1965

Extensions of matroids to se ts containing one additional ele ment are c ha rac te ri zed in te rm s or mod ular cuts of the latti ce or closed s ubse ts. An equivalent charac teri za ti o n is give…

### A catalogue of combinatorial geometries

- Mathematics
- 1973

We include in the microfiche section of this issue a catalogue of all the different geometric configurations which may be constructed from n points, n _ 8, by specifying their points, lines, planes,…

### Matroid Enumeration for Incidence Geometry

- MathematicsDiscret. Comput. Geom.
- 2012

A new algorithm for the enumeration of non-isomorphic matroids is developed, using an encoding of oriented matroid axioms as a boolean satisfiability (SAT) problem, which succeeds to enumerate a complete list of the isomorph-free rank 4 matroIDS on 10 elements.

### Matroids, trees, stable sets

- Mathematics
- 2003

Introduction.- Paths and Flows.- Bipartite Matching and Covering.- Nonbipartite Matching and Covering.- Matroids and Submodular Functions.- Trees, Branchings, and Connectors.- Cliques, Stable Sets…

### McKay ’ s Canonical Graph Labeling Algorithm

- Mathematics, Computer Science
- 2008

This article aims to provide an introduction to the essential ideas of McKay’s algorithm, the most powerful and best known of these algorithms for graph isomorphism, which has exponential running time on some inputs, but performs exceptionally well under most circumstances.

### Combinatorial optimization. Polyhedra and efficiency.

- Economics
- 2003

This book shows the combinatorial optimization polyhedra and efficiency as your friend in spending the time in reading a book.