Nick Brettell
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Nick Brettell has a total of 55 co-authors. Here are the top co-authors by number of shared publications:
- Daniël Paulusma
Durham University, Durham, UK
17 shared publications - Charles Semple
School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand
11 shared publications - Geoff Whittle
School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
9 shared publications - Andrea Munaro
Department of Mathematical, Physical and Computer Sciences University of Parma Italy
7 shared publications Geoff Whittle
School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand
7 shared publicationsCharles Semple
School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand
6 shared publications
Latest Publications
- 12 February 2026
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Discrete Mathematics
- Vol. 40 (1) , 134-169
Clonal Cores and Flexipaths in Matroids
A partitioned matroid [Formula: see text] consists of a matroid [Formula: see text] and a partition [Formula: see text] of its ground set. As such structures arise frequently in structural matroid theory, this paper introduces a general technique for analyzing those special properties of partitioned matroids that depend solely on the values of the connectivities [Formula: see text], the local connectivities [Formula: see text], and the dual local connectivities [Formula: see text]. In particular, we consider those partitioned matroids in which each [Formula: see text] is an independent, coindependent set of clones of cardinality [Formula: see text]. Calling such partitioned matroids clonal-core matroids, we show that special results of the above type for partitioned matroids can be verified in general by proving them just for clonal-core matroids. Aiming at the long-term goal of finding the unavoidable minors of 4-connected matroids, we illustrate this technique by studying 4-paths. These are sequences [Formula: see text] of sets that partition the ground set of a matroid so that the union of any proper initial segment of parts is 4-separating. Viewing the ends [Formula: see text] and [Formula: see text] as fixed, we call such a partition a 4-flexipath if [Formula: see text] is a 4-path for all permutations [Formula: see text] of [Formula: see text]. A straightforward simplification enables us to focus on [Formula: see text]-flexipaths for some [Formula: see text] in [Formula: see text], that is, those 4-flexipaths for which [Formula: see text] and [Formula: see text] for all distinct [Formula: see text] and [Formula: see text]. Our main result for 4-paths is that the only nontrivial case that arises here is when [Formula: see text]. In that case, there are essentially only two possible dual pairs of [Formula: see text]-flexipaths when [Formula: see text].
- 12 February 2026
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Discrete Mathematics
- Vol. 40 (1) , 134-169
Clonal Cores and Flexipaths in Matroids
A partitioned matroid [Formula: see text] consists of a matroid [Formula: see text] and a partition [Formula: see text] of its ground set. As such structures arise frequently in structural matroid theory, this paper introduces a general technique for analyzing those special properties of partitioned matroids that depend solely on the values of the connectivities [Formula: see text], the local connectivities [Formula: see text], and the dual local connectivities [Formula: see text]. In particular, we consider those partitioned matroids in which each [Formula: see text] is an independent, coindependent set of clones of cardinality [Formula: see text]. Calling such partitioned matroids clonal-core matroids, we show that special results of the above type for partitioned matroids can be verified in general by proving them just for clonal-core matroids. Aiming at the long-term goal of finding the unavoidable minors of 4-connected matroids, we illustrate this technique by studying 4-paths. These are sequences [Formula: see text] of sets that partition the ground set of a matroid so that the union of any proper initial segment of parts is 4-separating. Viewing the ends [Formula: see text] and [Formula: see text] as fixed, we call such a partition a 4-flexipath if [Formula: see text] is a 4-path for all permutations [Formula: see text] of [Formula: see text]. A straightforward simplification enables us to focus on [Formula: see text]-flexipaths for some [Formula: see text] in [Formula: see text], that is, those 4-flexipaths for which [Formula: see text] and [Formula: see text] for all distinct [Formula: see text] and [Formula: see text]. Our main result for 4-paths is that the only nontrivial case that arises here is when [Formula: see text]. In that case, there are essentially only two possible dual pairs of [Formula: see text]-flexipaths when [Formula: see text].
- 2 January 2026
- Published by Springer Nature
- p. 121-134
Non-crossing H-Graphs: A Generalization of Proper Interval Graphs Admitting FPT Algorithms
- 2 January 2026
- Published by Springer Nature
- p. 121-134
Non-crossing H-Graphs: A Generalization of Proper Interval Graphs Admitting FPT Algorithms
- Open Access
The excluded minors of the class of spike minors
Mayhew et al.\ (2021) posed the problem of showing that the minor-closed class of spikes and their minors has a finite set of excluded minors and describing all of them. In this paper, we resolve this problem.
Open AccessThe excluded minors of the class of spike minors
Mayhew et al.\ (2021) posed the problem of showing that the minor-closed class of spikes and their minors has a finite set of excluded minors and describing all of them. In this paper, we resolve this problem.
- Open Access
- 23 September 2025
- Published by Elsevier in Journal of Combinatorial Theory, Series B
- Vol. 176, 163-240
Detachable pairs in 3-connected matroids and simple 3-connected graphs
- 23 September 2025
- Published by Elsevier in Journal of Combinatorial Theory, Series B
- Vol. 176, 163-240
Open AccessDetachable pairs in 3-connected matroids and simple 3-connected graphs
- Open Access
- 3 May 2025
- Published by Elsevier in European Journal of Combinatorics
- Vol. 127
Comparing width parameters on graph classes
- 3 May 2025
- Published by Elsevier in European Journal of Combinatorics
- Vol. 127
Open AccessComparing width parameters on graph classes
- Open Access
- 11 April 2025
- Published by The Electronic Journal of Combinatorics in The Electronic Journal of Combinatorics
- Vol. 32 (2) , P2.1
What is a $4$-Connected Matroid?
The breadth of a tangle $\mathcal{T}$ in a matroid is the size of the largest spanning uniform submatroid of the tangle matroid of $\mathcal{T}$. The matroid $M$ is weakly $4$-connected if it is 3-connected and whenever $(X,Y)$ is a partition of $E(M)$ with $|X|,|Y|>4$, then $\lambda(X)\geq 3$. We prove that if $\mathcal{T}$ is a tangle of order $k\geq 4$ and breadth $l$ in a matroid $M$, then $M$ has a weakly 4-connected minor $N$ with a tangle $\mathcal{T}_N$ of order $k$, breadth $l$ and has the property that $\mathcal{T}$ is the tangle in $M$ induced by $\mathcal{T}_N$. A set $Z$ of elements of a matroid $M$ is $4$-connected if $\lambda(A)\geq\min\{|A\cap Z|,|Z-A|,3\}$ for all $A\subseteq E(M)$. As a corollary of our theorems on tangles we prove that if $M$ contains an $n$-element $4$-connected set where $n\geq 7$, then $M$ has a weakly $4$-connected minor that contains an $n$-element $4$-connected set.- 11 April 2025
- Published by The Electronic Journal of Combinatorics in The Electronic Journal of Combinatorics
- Vol. 32 (2) , P2.1
Open AccessWhat is a $4$-Connected Matroid?
The breadth of a tangle $\mathcal{T}$ in a matroid is the size of the largest spanning uniform submatroid of the tangle matroid of $\mathcal{T}$. The matroid $M$ is weakly $4$-connected if it is 3-connected and whenever $(X,Y)$ is a partition of $E(M)$ with $|X|,|Y|>4$, then $\lambda(X)\geq 3$. We prove that if $\mathcal{T}$ is a tangle of order $k\geq 4$ and breadth $l$ in a matroid $M$, then $M$ has a weakly 4-connected minor $N$ with a tangle $\mathcal{T}_N$ of order $k$, breadth $l$ and has the property that $\mathcal{T}$ is the tangle in $M$ induced by $\mathcal{T}_N$. A set $Z$ of elements of a matroid $M$ is $4$-connected if $\lambda(A)\geq\min\{|A\cap Z|,|Z-A|,3\}$ for all $A\subseteq E(M)$. As a corollary of our theorems on tangles we prove that if $M$ contains an $n$-element $4$-connected set where $n\geq 7$, then $M$ has a weakly $4$-connected minor that contains an $n$-element $4$-connected set. - Open Access
- 19 February 2025
- Published by Elsevier in Advances in Applied Mathematics
- Vol. 166
The excluded minors for GF(5)-representable matroids on ten elements
- 19 February 2025
- Published by Elsevier in Advances in Applied Mathematics
- Vol. 166
Open AccessThe excluded minors for GF(5)-representable matroids on ten elements
- Open Access
- 28 January 2025
- Published by Elsevier in Theoretical Computer Science
- Vol. 1032
Computing subset vertex covers in H-free graphs
- 28 January 2025
- Published by Elsevier in Theoretical Computer Science
- Vol. 1032
Open AccessComputing subset vertex covers in H-free graphs
- Open Access
- 12 July 2024
- Published by The Electronic Journal of Combinatorics in The Electronic Journal of Combinatorics
- Vol. 31 (3) , P3.20
Computing Excluded Minors for Classes of Matroids Representable over Partial Fields
We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids.
- 12 July 2024
- Published by The Electronic Journal of Combinatorics in The Electronic Journal of Combinatorics
- Vol. 31 (3) , P3.20
Open AccessComputing Excluded Minors for Classes of Matroids Representable over Partial Fields
We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids.
- Open Access
- 7 November 2023
- Published by Elsevier in Journal of Computer and System Sciences
- Vol. 140
Solving problems on generalized convex graphs via mim-width
- 7 November 2023
- Published by Elsevier in Journal of Computer and System Sciences
- Vol. 140
Open AccessSolving problems on generalized convex graphs via mim-width