Daryl Funk
YOU?
Author Swipe
View article: There are only a finite number of excluded minors for the class of bicircular matroids
There are only a finite number of excluded minors for the class of bicircular matroids Open
We show that the class of bicircular matroids has only a finite number of excluded minors. Key tools used in our proof include representations of matroids by biased graphs and the recently introduced class of quasi-graphic matroids. We sho…
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: 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: Short rainbow cycles in graphs and matroids
Short rainbow cycles in graphs and matroids Open
Let be a simple ‐vertex graph and be a coloring of with colors, where each color class has size at least 2. We prove that contains a rainbow cycle of length at most , which is best possible. Our result settles a special case of a strengthe…
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: 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: Matrix representations of frame and lifted-graphic matroids correspond to gain functions
Matrix representations of frame and lifted-graphic matroids correspond to gain functions Open
Let $M$ be a 3-connected matroid and let $\mathbb F$ be a field. Let $A$ be a matrix over $\mathbb F$ representing $M$ and let $(G,\mathcal B)$ be a biased graph representing $M$. We characterize the relationship between $A$ and $(G,\mathc…
View article: Matrix representations of matroids of biased graphs correspond to gain functions
Matrix representations of matroids of biased graphs correspond to gain functions Open
Let $M$ be a frame matroid or a lifted-graphic matroid and let $(G,\mathcal{B})$ be a biased graph representing $M$. Given a field $\mathbb{F}$, a canonical $\mathbb{F}$-representation of $M$ particular to $(G,\mathcal{B})$ is a matrix $A$…