Related papers: Twist monomials of binary delta-matroids
A mixed graph is a graph with some directed edges and some undirected edges. We introduce the notion of mixed matroids as a generalization of mixed graphs. A mixed matroid can be viewed as an oriented matroid in which the signs over a fixed…
In this paper, we survey results regarding the interlace polynomial of a graph, connections to such graph polynomials as the Martin and Tutte polynomials, and generalizations to the realms of isotropic systems and delta-matroids.
A classic exercise in the topology of surfaces is to show that, using handle slides, every disc-band surface, or 1-vertex ribbon graph, can be put in a canonical form consisting of the connected sum of orientable loops, and either…
Given a matroid or flag of matroids we introduce several broad classes of polynomials satisfying Deletion-Contraction identities, and study their singularities. There are three main families of polynomials captured by our approach:…
We prove that the extension complexity of the independence polytope of every regular matroid on $n$ elements is $O(n^6)$. Past results of Wong and Martin on extended formulations of the spanning tree polytope of a graph imply a $O(n^2)$…
Given a map $\mathcal M$ on a connected and closed orientable surface, the delta-matroid of $\mathcal M$ is a combinatorial object associated to $\mathcal M$ which captures some topological information of the embedding. We explore how…
Twisted graph diagrams are virtual graph diagrams with bars on edges. A bijection between abstract graph diagrams and twisted graph diagrams is constructed. Then a polynomial invariant of Yamada-type is developed which provides a lower…
Delta-matroids are "type B" generalizations of matroids in the same way that maximal orthogonal Grassmannians are generalizations of Grassmannians. A delta-matroid analogue of the Tutte polynomial of a matroid is the interlace polynomial.…
In contrast to matroids, vf-safe delta-matroids have three kinds of minors and are closed under the operations of twist and loop complementation. We show that the delta-matroids representable over GF(4) with respect to the nontrivial…
A transversal matroid whose dual is also transversal is called bi-transversal. Let $G$ be an undirected graph with vertex set $V$. In this paper, for every subset $W$ of $V$, we associate a bi-transversal matroid to the pair $(G,W)$. We…
We introduce a binary matroid M(IAS(G)) associated with a looped simple graph G. M(IAS(G)) classifies G up to local equivalence, and determines the delta-matroid and isotropic system associated with G. Moreover, a parametrized form of its…
We provide a characterisation of when a single-element contraction of a transversal matroid is itself transversal. Using this characterisation, we define a new class of transversal matroids closed under minors, which we call path-circular…
We characterize the shifted simple graphs and the $3$-uniform shifted hypergraphs whose inverse image under exterior shifting is the set of bases of a matroid: those are exactly the hypergraphs whose hyperedges form an initial lex-segment.…
We primarily investigate the properties of characteristic polynomials of semimatroids. In particular, we provide a combinatorial interpretation of their coefficients, generalizing the Whitney's Broken Circuit Theorem. We also prove that the…
In this work, we study the operations of handle slides introduced recently for delta-matroids by Iain Moffatt and Eunice Mphako-Banda. We then prove that the class of binary delta-matroids is the only class of delta-matroids closed under…
We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang…
This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a)…
We present a simplification of the recursive algorithm for the evaluation of intersection numbers for differential $n$-forms, by combining the advantages emerging from the choice of delta-forms as generators of relative twisted cohomology…
We characterize the class of threshold matroids by the structure of their defining bases. We also give an example of a shifted matroid which is not threshold, answering a question of Deza and Onn. We conclude by exploring consequences of…
We provide a combinatorial study of split matroids, a class that was motivated by the study of matroid polytopes from a tropical geometry point of view. A nice feature of split matroids is that they generalize paving matroids, while being…