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View article: Dominic Welsh: his work and influence
Dominic Welsh: his work and influence Open
We review the work of Dominic Welsh (1938-2023), tracing his remarkable influence through his theorems, expository writing, students, and interactions. He was particularly adept at bringing different fields together and fostering the devel…
View article: Reduced Clique Graphs: A Correction to “Chordal Graphs and Their Clique Graphs”
Reduced Clique Graphs: A Correction to “Chordal Graphs and Their Clique Graphs” Open
Galinier, Habib, and Paul introduced the reduced clique graph of a chordal graph G . The nodes of the reduced clique graph are the maximal cliques of G , and two nodes are joined by an edge if and only if they form a non-disjoint separatin…
View article: Monadic transductions and definable classes of matroids
Monadic transductions and definable classes of matroids Open
A transduction provides us with a way of using the monadic second-order language of a structure to make statements about a derived structure. Any transduction induces a relation on the set of these structures. This article presents a self-…
View article: Tree Automata and Pigeonhole Classes of Matroids: II
Tree Automata and Pigeonhole Classes of Matroids: II Open
Let $\psi$ be a sentence in the counting monadic second-order logic of matroids and let F be a finite field. Hlineny's Theorem says that we can test whether F-representable matroids satisfy $\psi$ using an algorithm that is fixed-parameter…
View article: Reduced clique graphs: a correction to "Chordal graphs and their clique graphs"
Reduced clique graphs: a correction to "Chordal graphs and their clique graphs" Open
Galinier, Habib, and Paul introduced the reduced clique graph of a chordal graph $G$. The nodes of the reduced clique graph are the maximal cliques of $G$, and two nodes are joined by an edge if and only if they form a non-disjoint separat…
View article: Supersolvable saturated matroids and chordal graphs
Supersolvable saturated matroids and chordal graphs Open
A matroid is supersolvable if it has a maximal chain of flats each of which is modular. A matroid is saturated if every round flat is modular. In this article we present supersolvable saturated matroids as analogues to chordal graphs, and …
View article: Matroids with 9 elements
Matroids with 9 elements Open
This dataset contains the 385370 pairwise non-isomorphic matroids from 0 to 9 elements inclusive. Each matroid is given as a text string occupying one line of the file matroids09_rankLine. Except for the empty matroid, the text string give…
View article: Matroids with 9 elements
Matroids with 9 elements Open
This dataset contains the 385370 pairwise non-isomorphic matroids from 0 to 9 elements inclusive. Each matroid is given as a text string occupying one line of the file matroids09_rankLine. Except for the empty matroid, the text string give…
View article: Tree Automata and Pigeonhole Classes of Matroids: I
Tree Automata and Pigeonhole Classes of Matroids: I Open
Hliněný’s Theorem shows that any sentence in the monadic second-order logic of matroids can be tested in polynomial time, when the input is limited to a class of $${\mathbb {F}}$$ -representable matroids with bounded branch-width (where …
View article: Defining Bicircular Matroids in Monadic Logic
Defining Bicircular Matroids in Monadic Logic Open
We conjecture that the class of frame matroids can be characterized by a sentence in the monadic second-order logic of matroids, and we prove that there is such a characterization for the class of bicircular matroids. The proof does not de…
View article: On the relative importance of excluded minors
On the relative importance of excluded minors Open
If E is a set of matroids, then EX(E) denotes the set of matroids that have no minor isomorphic to a member of E. If E′ E, we say that E′ is superfluous if EX(E-E′)-EX(E) contains only finitely many 3-connected matroids. We determine the s…
View article: Obstructions for bounded branch-depth in matroids
Obstructions for bounded branch-depth in matroids Open
DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid $U_{n,2n}$ or the cycle …
View article: Excluded minors for the class of split matroids
Excluded minors for the class of split matroids Open
Split matroids form a minor-closed class of matroids, and are defined by placing conditions on the system of split hyperplanes in the matroid base polytope. They can equivalently be defined in terms of structural properties involving cycli…
View article: Excluding Kuratowski graphs and their duals from binary matroids
Excluding Kuratowski graphs and their duals from binary matroids Open
© 2017 Elsevier Inc. We consider some applications of our characterisation of the internally 4-connected binary matroids with no M(K3,3)-minor. We characterise the internally 4-connected binary matroids with no minor in M, where M is a sub…
View article: Excluding Kuratowski graphs and their duals from binary matroids
Excluding Kuratowski graphs and their duals from binary matroids Open
© 2017 Elsevier Inc. We consider some applications of our characterisation of the internally 4-connected binary matroids with no M(K3,3)-minor. We characterise the internally 4-connected binary matroids with no minor in M, where M is a sub…
View article: Effective Versions of Two Theorems of RADO
Effective Versions of Two Theorems of RADO Open
Let $M$ be a representable matroid on $n$ elements. We give bounds, in terms of $n$, on the least positive characteristic and smallest field over which $M$ is representable.
View article: Tree automata and pigeonhole classes of matroids: I
Tree automata and pigeonhole classes of matroids: I Open
Hlineny's Theorem shows that any sentence in the monadic second-order logic of matroids can be tested in polynomial time, when the input is limited to a class of F-representable matroids with bounded branch-width (where F is a finite field…
View article: Tree automata and pigeonhole classes of matroids: II
Tree automata and pigeonhole classes of matroids: II Open
Let $ψ$ be a sentence in the counting monadic second-order logic of matroids and let $\mathbb{F}$ be a finite field. Hliněný's Theorem says that we can test whether $\mathbb{F}$-representable matroids satisfy $ψ$ using an algorithm that is…
View article: Structure of Cubic Lehman Matrices
Structure of Cubic Lehman Matrices Open
A pair $(A,B)$ of square $(0,1)$-matrices is called a Lehman pair if $AB^T=J+kI$ for some integer $k\in\{-1,1,2,3,\ldots\}$. In this case $A$ and $B$ are called Lehman matrices. This terminology arises because Lehman showed that the rows w…
View article: Well-quasi-ordering in lattice path matroids
Well-quasi-ordering in lattice path matroids Open
Lattice path matroids form a subclass of transversal matroids and were introduced by Bonin, de Mier and Noy. Transversal matroids are not well-quasi-ordered, even when the branch-width is restricted. Though lattice path matroids are not we…
View article: Excluded minors for the class of split matroids
Excluded minors for the class of split matroids Open
Split matroids form a minor-closed class of matroids, and are defined by placing conditions on the system of split hyperplanes in the matroid base polytope. They can equivalently be defined in terms of structural properties involving cycli…
View article: On excluded minors for classes of graphical matroids
On excluded minors for classes of graphical matroids Open
Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle…
View article: A splitter theorem for connected clutters
A splitter theorem for connected clutters Open
A clutter consists of a finite set and a collection of pairwise incomparable subsets. Clutters are natural generalisations of matroids, and they have similar operations of deletion and contraction. We introduce a notion of connectivity for…
View article: The antichain of excluded minors for the class of gammoids is maximal
The antichain of excluded minors for the class of gammoids is maximal Open
Every gammoid is a minor of an excluded minor for the class of gammoids.