## Research interests:

Matroid theory and graph theory. The paper “What is a matroid?” provides an introduction to matroid theory. This paper
is a revision of a paper that appeared in *Cubo* **5** (2003), 179–218.
An even shorter introduction to matroid theory can be found in the paper “Briefly, what is a matroid?”.

A second edition of my book [**Matroid Theory, Oxford University Press, New York**] was published on February 17, 2011 in the UK,
and in April, 2011 in the USA. This is a major revision of the book.

- Hardback and paperback of the second edition on the Oxford University Press website.
- Errata and update on conjectures, problems, and references from the second edition (2011) of the book.

Errata and update on conjectures, problems, and references from the first edition (1992) of the book. These changes were incorporated into a July 2006 reprint of the first edition.

## Teaching:

### Fall, 2022.

**Math 7001-01. Communicating Mathematics I.** TuTh 3:00 – 4:50.

- Halmos's article on how to write mathematics [pdf].
- Guide for writing a teaching statement [pdf].

**Office Hours:** Fall, 2022. By appointment.

## Recent publications:

Preprints of all of my papers since 2000 and certain other earlier papers may be obtained by clicking on the paper titles below.

- (with G. Farr and D. Mayhew) Dominic Welsh: his work and influence, The Matroid Union blog, Posted February 12, 2024.
- Donald Hawkins Row, August 26, 1939 - March 29, 2023, submitted.
- (with N. Brettell, S. Jowett, C. Semple, and G. Whittle) What is a 4-connected matroid?, submitted.
- (with M. Mizell) Matroids arising from nested sequences of flats in projective and affine geometries, submitted.
- (with J.D. Douthitt)
Chordal matroids arising from generalized parallel connections,
*Adv. in Appl. Math.***153**(2024), Paper 102631, 10pp. - (with N. Brettell, C. Semple, and G. Whittle)
Excluded minors are almost fragile II: essential elements,
*J. Combin. Theory Ser. B***163**(2023), 272–307. - (with N. Brettell, C. Semple, and G. Whittle)
The excluded minors for 2- and 3-regular matroids,
*J. Combin. Theory Ser. B***163**(2023), 133–218. - (with C. Crenshaw)
A binary-matroid analogue of a graph connectivity theorem of Jamison and Mulder,
*Adv. in Appl. Math.***151**(2023), 102577, 12pp. - (with C. Crenshaw)
On the cogirth of binary matroids,
*Adv. in Appl. Math.***147**(2023), Paper 102515, 9pp. - (with M. Lemos)
An upper bound for the circumference of a 3-connected binary matroid,
*Electron. J. Combin.***30**(2023), Paper 1.45, 12pp. - (with J. Singh)
Generalizing cographs to 2-cographs,
*Electron. J. Combin.***30**(2023), Paper 1.1, 32pp. - Odd circuits in binary matroids, The Matroid Union blog, Posted May 3, 2022.
- (with C. Crenshaw)
Ordering circuits of matroids,
*Electron. J. Combin.***29**(2022), Paper 4.31, 28pp. - (with J. Singh)
The smallest classes of binary and ternary matroids closed under direct sums and complements,
*SIAM J. Discrete Math.***36**(2022), 2051–2072. - (with Z. Walsh)
Small cocircuits in minimally vertically 4-connected matroids,
*SIAM J. Discrete Math.***36**(2022), 2822–2829. - (with S. Pfeil)
Unbreakable matroids,
*Adv. in Appl. Math.***141**(2022), Paper 102404, 7pp. - (with Z. Walsh)
2-modular matrices,
*SIAM J. Discrete Math.***36**(2022), 1231–1248. - (with J.D. Douthitt)
Bounding the number of bases of a matroid,
*Discrete Math.***345**(2022), 112636, 3pp. - (with C. Chun and K. Wetzler)
The binary matroids with no odd circuits of size exceeding five,
*J. Combin. Theory Ser. B***152**(2022), 80–120. - (with B. Clark, K. Grace, and S. H. M. van Zwam)
On the highly connected dyadic, near-regular, and sixth-root-of-unity matroids,
*SIAM J. Discrete Math.***35**(2021), 1356–1380. - (with J. Singh)
Complementation, local complementation, and switching in binary matroids,
*Adv. in Appl. Math.***126**(2021), Paper 102041, 13pp. - (with G. Drummond, T. Fife, and K. Grace)
Circuit-difference matroids,
*Electron. J. Combin.***27**(2020), Paper 3.11, 11pp. - (with S. Wang) A variant on the circuit elimination axiom.
- (with T. Fife, D. Mayhew, and C. Semple)
The unbreakable frame matroids,
*SIAM J. Discrete Math.***34**(2020), 1522–1537. - (with N. Brettell, B. Clark, C. Semple, and G. Whittle)
Excluded minors are almost fragile,
*J. Combin. Theory Ser. B***140**(2020), 263–322. - (with R. Campbell, K. Grace, and G. Whittle)
On density-critical matroids,
*Electron. J. Combin.***27**(2020), Paper 2.35, 16pp. - Henry Crapo: A brief reminiscence, The Matroid Union blog, Posted October 21, 2019.
- (with S. Wang)
Dependencies among dependencies in matroids,
*Electron. J. Combin.***26**(2019), Paper 3.46, 12pp. - (with Z. Gershkoff)
A note on the connectivity of 2-polymatroid minors,
*Electron. J. Combin.***26**(2019), Paper 4.21, 11pp. - (with C. Semple and G. Whittle)
A Splitter Theorem for 3-connected 2-polymatroids,
*Electron. J. Combin.***26**(2019), Paper 2.37, 95pp. - (with G. Farr)
The contributions of W.T. Tutte to matroid theory,
*2017 MATRIX Annals*(D. Wood editor-in-chief), Springer, 2019, pp. 343-361. - A matroid extension result,
*SIAM J. Discrete Math.***33**(2019), 138-152. - (with S. Pfeil, C. Semple, and G. Whittle)
Matroids with many small circuits and cocircuits,
*Adv. in Appl. Math.***105**(2019), 1–24. - (with T. Fife)
Generalized laminar matroids,
*Europ. J. Combin.***79**(2019), 111–122. - (with C. Chun)
Towards a Splitter Theorem for internally 4-connected binary matroids VII,
*Adv. in Appl. Math.***104**(2019), 14–74. - (with C. Chun)
Internally 4-connected binary matroids with every element in three triangles,
*Combinatorica***39**(2019), 825–845. - (with Z. Gershkoff)
A notion of minor-based matroid connectivity,
*Adv. in Appl. Math.***100**(2018), 163–178. - (with B. Clark and S.H.M. van Zwam)
Relaxations of GF(4)-representable matroids,
*Electron. J. Combin.***25**(2018), Paper 2.53, 23pp. - (with C. Chun)
Towards a Splitter Theorem for internally 4-connected binary matroids VI,
*Adv. in Appl. Math.***101**(2018), 100–167. - (with T. Fife)
Laminar matroids,
*Europ. J. Combin.***62**(2017), 206–216. - (with C. Chun and D. Mayhew)
Towards a Splitter Theorem for internally 4-connected binary matroids VIII: small matroids,
*Adv. in Appl. Math.***85**(2017), 12–30. - (with C. Chun and D. Mayhew)
Towards a Splitter Theorem for internally 4-connected binary matroids IX: The theorem,
*J. Combin. Theory Ser. B***121**(2016), 2–67. - A matroid analogue of a theorem of Brooks for graphs,
*Europ. J. Combin.***53**(2016), 45–49. - (with C. Semple and G. Whittle)
Determining a binary matroid from its small circuits,
*Electron. J. Combin.***23**(2016), Paper 1.26, 7pp. - (with K. Wetzler)
The binary matroids whose only odd circuits are triangles,
*Adv. in Appl. Math.***76**(2016), 34–38. - (with C. Chun, G. Ding, and D. Mayhew)
Unavoidable connected matroids retaining a specified minor,
*SIAM J. Discrete Math.***30**(2016), 1590–1606. - (with C. Semple and G. Whittle)
A Wheels-and-Whirls Theorem for 3-connected 2-polymatroids,
*SIAM J. Discrete Math.***30**(2016), 493–524. - (with C. Chun and D. Mayhew)
Towards a Splitter Theorem for internally 4-connected binary matroids V,
*Adv. in Appl. Math.***52**(2014), 60–81. - (with C. Chun and D. Mayhew)
Towards a Splitter Theorem for internally 4-connected binary matroids IV,
*Adv. in Appl. Math.***52**(2014), 1–59. - (with J. Taylor)
On two classes of nearly binary matroids,
*Europ. J. Combin.***36**(2014), 251–260. - (with C. Chun and D. Mayhew)
Towards a Splitter Theorem for internally 4-connected binary matroids II,
*Europ. J. Combin.***36**(2014), 550–563. - (with C. Chun and D. Mayhew)
Towards a Splitter Theorem for internally 4-connected binary matroids III,
*Adv. in Appl. Math.***51**(2013), 309–344. - (with D. Chun)
Capturing two elements in unavoidable minors of 3-connected binary matroids,
*Adv. in Appl. Math.***50**(2013), 155–175. - (with C. Chun and D. Mayhew)
Constructing internally 4-connected binary matroids,
*Adv. in Appl. Math.***50**(2013), 16–45. - (with C. Semple)
Constructing a 3-tree for a 3-connected matroid,
*Adv. in Appl. Math.***50**(2013), 176–200. - (with L. Lowrance, C. Semple, and D. Welsh)
On properties of almost all matroids,
*Adv. in Appl. Math.***50**(2013), 115–124. - (with C. Chun and D. Mayhew)
Towards a splitter theorem for internally 4-connected binary matroids,
*J. Combin. Theory Ser. B***102**(2012), 688–700. - (with D. Chun and G. Whittle)
Capturing matroid elements in unavoidable 3-connected minors,
*Europ. J. Combin.***33**(2012), 1100–1112. - (with J. Aikin)
The structure of the 4-separations in 4-connected matroids,
*Adv. in Appl. Math.***48**(2012), 1–24. - (with C. Semple and G. Whittle)
An upgraded Wheels-and-Whirls Theorem for 3-connected matroids,
*J. Combin. Theory Ser. B***102**(2012), 610–637. - On bipartite restrictions of binary matroids,
*Europ. J. Combin.***32**(2011), 1199–1202. - (with C. Semple and G. Whittle)
Exposing 3-separations in 3-connected matroids,
*Adv. in Appl. Math.***47**(2011), 463–508. - (with C. Chun and D. Mayhew)
A chain theorem for internally 4-connected binary matroids,
*J. Combin. Theory Ser. B***101**(2011), 141–189. - (with C. Chun)
Unavoidable parallel minors of regular matroids,
*Europ. J. Combin.***32**(2011), 762–774. - (with D. Mayhew, B. Oporowski, and G. Whittle)
The excluded minors for the matroids that are binary or ternary,
*Europ. J. Combin.***32**(2011), 891–930. - (with N. Hine)
When excluding one matroid prevents infinite antichains,
*Adv. in Appl. Math.***45**(2010), 74–76. - (with J. Aikin)
The structure of crossing separations in matroids,
*Adv. in Appl. Math.***41**(2008), 10–26. - (with C. Semple and G. Whittle)
Maintaining 3-connectivity relative to a fixed basis,
*Adv. in Appl. Math.***41**(2008), 1–9. - (with B. Beavers)
Constructive characterizations of 3-connected matroids of path width three,
*Europ. J. Combin.***29**(2008), 1643–1661. - (with C. Semple and G. Whittle)
A chain theorem for matroids,
*J. Combin. Theory Ser. B***98**(2008), 447–483. - (with C. Semple and G. Whittle)
Wild triangles in 3-connected matroids,
*J. Combin. Theory Ser. B***98**(2008), 291–323. - (with C. Semple and G. Whittle)
The structure of the 3-separations of 3-connected matroids II,
*Europ. J. Combin.***28**(2007), 1239–1261. - (with R. Hall and C. Semple)
The structure of 3-connected matroids of path width three,
*Europ. J. Combin.***28**(2007), 964–989. - The contributions of Dominic Welsh to matroid theory,
in
*Combinatorics, Complexity, and Chance*(G. Grimmett and C. McDiarmid eds.), Oxford Univ. Press, Oxford, 2007, pp.234–259. - (with M. Lemos)
Matroid covering and packing with circuits through an element,
*J. Combin. Theory Ser. B***96**(2006), 135–158. - (with B. Beavers)
On pancyclic representable matroids,
*Discrete Math.***305**(2005), 337–343. - (with R. Hall and C. Semple)
The structure of equivalent 3-separations in a 3-connected matroid,
*Adv. in Appl. Math.***35**(2005), 123–181. - (with H. Wu)
The 3-connected graphs with exactly three non-essential edges,
*Graphs Combin.***20**(2004), 233–246. - (with C. Semple and G. Whittle)
The structure of the 3-separations of 3-connected matroids,
*J. Combin. Theory Ser. B***92**(2004), 257–293. - (with J.F. Geelen, D.L. Vertigan, and G.P. Whittle)
A short proof of non-GF(5)-representability of matroids,
*J. Combin. Theory Ser. B***91**(2004), 105–121. - (with R. Hall, C. Semple, and G. Whittle)
Fork-decompositions of matroids,
*Adv. in Appl. Math.***32**(2004), 523–575. - (with M. Lemos)
On the minor-minimal 2-connected graphs having a fixed minor,
*Discrete Math.***280**(2004), 77–118. - (with A.M. Hobbs)
William T. Tutte, 1917–2002,
*Notices Amer. Math. Soc.***51**(2004), 320–330. - (with Y.-B. Choe, A.D. Sokal, and D.G. Wagner)
Homogeneous multivariate polynomials with the half-plane property,
*Adv. in Appl. Math.***32**(2004), 88–187. - (with M. Lemos)
On the minor-minimal 3-connected matroids having a fixed minor,
*Europ. J. Combin.***24**(2003), 1097–1123. - (with B. Chaourar)
On series-parallel extensions of uniform matroids,
*Europ. J. Combin.***24**(2003), 877–879. - The structure of a 3-connected matroid with a 3-separating set of essential elements,
*Discrete Math.***265**(2003), 173–187. - (with M. Lemos)
On removable cycles through every edge,
*J. Graph Theory***42**(2003), 155–164. - (with R. Hall, C. Semple, and G. Whittle)
On matroids of branch-width three,
*J. Combin. Theory Ser. B***86**(2002), 148–171. - (with D. Welsh)
Chromatic, flow, and reliability polynomials: the complexity of their coefficients,
*Combinatorics, Probability and Computing***11**(2002), 403–426. - (with J.F. Geelen, D.L. Vertigan, and G.P. Whittle)
Totally free expansions of matroids,
*J. Combin. Theory Ser. B***84**(2002), 130–179. - (with C. Semple, D. Vertigan, and G. Whittle)
Infinite antichains of matroids with characteristic set
*p*,*Discrete Math.***242**(2002), 175–185. - On the interplay between graphs and matroids,
*Surveys in Combinatorics, 2001 (Sussex)*(J.W.P. Hirschfeld ed.) London Math. Soc. Lecture Notes**288**, Cambridge Univ. Press, Cambridge, 2001, pp. 199–239. - (with M. Lemos)
A sharp bound on the size of a connected matroid,
*Trans. Amer. Math. Soc.***353**(2001), 4039–4056. - (with J.F. Geelen, D.L. Vertigan, and G.P. Whittle)
On the excluded minors for quaternary matroids,
*J. Combin. Theory Ser. B***80**(2000), 57–68. - (with H. Wu)
On the structure of 3-connected matroids and graphs,
*Europ. J. Combin.***21**(2000), 667–688. - (with M. Lemos and T.J. Reid)
On the 3-connected matroids that are minimal having a fixed restriction,
*Graphs and Combin.***16**(2000), 285–318. - (with C. Semple and D. Vertigan)
Generalized Delta - Y exchanges and
*k*-regular matroids,*J. Combin. Theory Ser. B***79**(2000), 1–65. - (with M. Lemos)
On size, circumference and circuit removal in 3-connected matroids,
*Discrete Math.***220**(2000), 145–157. - (with H. Wu)
Matroids and graphs with few non-essential elements,
*Graphs and Combin.***16**(2000), 199–229. - (with G. Whittle)
On the non-uniqueness of
*q*-cones of matroids,*Discrete Math.***218**(2000), 271–275. - (with M. Lemos)
On the 3-connected matroids that are minimal having a fixed spanning restriction,
*Discrete Math.***218**(2000), 131–165. - A matroid generalization of a result of Dirac,
*Combinatorica***17**(1997), 267–273. - (with G. Ding, B. Oporowski, and D. Vertigan)
Unavoidable minors of large 3-connected matroids,
*J. Combin. Theory Ser. B***71**(1997), 244–293. - (with D. Vertigan and G. Whittle)
On inequivalent representations of matroids over finite fields,
*J. Combin. Theory Ser. B***67**(1996), 325–343. - (with G. Ding, B. Oporowski, and D. Vertigan)
Unavoidable minors of large 3-connected binary matroids,
*J. Combin. Theory Ser. B***66**(1996), 334–360. - Structure theory and connectivity for matroids,
*Matroid Theory*(J. Bonin, J. Oxley, and B. Servatius eds.) Contemporary Mathematics**197**(Amer. Math. Soc., Providence), (1996), pp. 129–170. - (with G. Ding and B. Oporowski)
On infinite antichains of matroids,
*J. Combin. Theory Ser. B***63**(1995), 21–40. - (with G. Whittle)
A characterization of Tutte invariants of 2-polymatroids,
*J. Combin. Theory Ser. B***59**(1993), 210–244. - (with B. Oporowski and R. Thomas)
Typical subgraphs of 3- and 4-connected graphs,
*J. Combin. Theory Ser. B***57**(1993), 239–257. - (with S. Akkari)
Some local extremal connectivity results for matroids,
*Combinatorics, Probability and Computing***2**(1993), 367–384. - (with T. Brylawski)
The Tutte polynomial and its applications,
in
*Matroid Applications*(N. White ed.), Cambridge Univ. Press, Cambridge, 1992, pp.123–225. - Infinite matroids,
in
*Matroid Applications*(N. White ed.), Cambridge Univ. Press, Cambridge, 1992, pp.73–90. - (with C.R. Coullard)
Extensions of Tutte's Wheels-and-Whirls Theorem,
*J. Combin. Theory Ser. B***56**(1992), 130–140. - Ternary paving matroids,
*Discrete Math.***91**(1991), 77–86. - (with S. Akkari)
Some extremal connectivity results for matroids,
*J. Combin. Theory Ser. B***52**(1991), 301–320. - (with G. Whittle)
A note on the non-spanning circuits of a matroid,
*Europ. J. Combin.***12**(1991), 259–261. - A characterization of a class of non-binary matroids,
*J. Combin. Theory Ser. B***49**(1990), 181–189. - The regular matroids with no 5-wheel minor,
*J. Combin. Theory Ser. B***46**(1989), 292–305. - The binary matroids with no 4-wheel minor,
*Trans. Amer. Math. Soc.***301**(1987), 63–75. - On non-binary 3-connected matroids,
*Trans. Amer. Math. Soc.***300**(1987), 663–679. - (with Y. Cheng)
On weakly symmetric graphs of order twice a prime,
*J. Combin. Theory Ser. B***42**(1987), 196–211. - A characterization of the ternary matroids with no M(K_4)-minor,
*J. Combin. Theory Ser. B***42**(1987), 212–249. - On the matroids representable over GF(4),
*J. Combin. Theory Ser. B***41**(1986), 250–252. - Proof of a conjecture of Kahn for non-binary matroids,
*Combinatorica***5**(1985), 343–345. - On the intersections of circuits and cocircuits in matroids,
*Combinatorica***4**(1984), 187–195. - (with D. Kelly)
On random representable matroids,
*Stud. Appl. Math.***71**(1984), 181–205. - On the numbers of bases and circuits in simple binary matroids,
*Europ. J. Combin.***4**(1983), 169–178. - (with K. Prendergast and D. Row)
Matroids whose ground sets are domains of functions,
*J. Austral. Math. Soc. (Series A)***32**(1982), 380–387. - On Crapo's beta invariant for matroids,
*Stud. Appl. Math.***66**(1982), 267–277. - On connectivity in matroids and graphs,
*Trans. Amer. Math. Soc.***265**(1981), 47–58. - On a matroid generalization of graph connectivity,
*Math. Proc. Camb. Phil. Soc.***90**(1981), 207–214. - On matroid connectivity,
*Quart. J. Math. Oxford Ser. 2***32**(1981), 193–208. - (with J.H. Mason)
A circuit covering result for matroids,
*Math. Proc. Camb. Phil. Soc.***87**(1980), 25–27. - Infinite matroids,
*Proc. London Math. Soc. (3)***37**(1978), 259–272. - Cocircuit coverings and packings for binary matroids,
*Math. Proc. Camb. Phil. Soc.***83**(1978), 347–351. - Colouring, packing and the critical problem,
*Quart. J. Math Oxford Ser. 2***29**(1978), 11–22.