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## ORBIT COUNTING AND THE TUTTE POLYNOMIAL

*Peter J. Cameron*

### in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0001
- Subject:
- Mathematics, Probability / Statistics

This chapter summarizes the various attempts to extend the Tutte polynomial of a matroid to a polynomial which counts orbits of a group on various sets of objects that the usual Tutte polynomial ... More

## THE CONTRIBUTIONS OF DOMINIC WELSH TO MATROID THEORY

*James Oxley*

### in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0015
- Subject:
- Mathematics, Probability / Statistics

Dominic Welsh began writing papers in matroid theory nearly forty years ago. Since then, he has made numerous important contributions to the subject. This chapter reviews Dominic Welsh's work in and ... More

## Graphic Matroids

*James Oxley*

### in Matroid Theory

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198566946
- eISBN:
- 9780191774904
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566946.003.0006
- Subject:
- Mathematics, Educational Mathematics

This chapter examines graphic matroids in more detail. In particular, it presents several proofs delayed from Chapters 1 and 2, including proofs that a graphic matroid is representable over every ... More

## Representable Matroids

*James Oxley*

### in Matroid Theory

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198566946
- eISBN:
- 9780191774904
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566946.003.0007
- Subject:
- Mathematics, Educational Mathematics

This chapter provides an overview of the basic questions associated with matroid representability and indicates how one actually goes about constructing representations. The key ideas are presented ... More

## Excluded-Minor Theorems

*James Oxley*

### in Matroid Theory

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198566946
- eISBN:
- 9780191774904
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566946.003.0011
- Subject:
- Mathematics, Educational Mathematics

This chapter is organized as follows. Section 10.1 presents Gerards' (1989) proof of Tutte's (1958) excluded-minor characterization of the class of regular matroids. Section 10.2 proves the ... More

## Submodular Functions and Matroid Union

*James Oxley*

### in Matroid Theory

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198566946
- eISBN:
- 9780191774904
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566946.003.0012
- Subject:
- Mathematics, Educational Mathematics

This chapter considers several more matroid constructions. The chapter is organized as follows. Section 11.2 considers several applications of submodular functions, one of which is in the proof of ... More

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