相关论文: Obstructions for matroids of path-width at most k …

We prove that for each finite field $\mathbb F$ and integer $k\in \mathbb Z$ there exists $n\in \mathbb Z$ such that no excluded minor for the class of $\mathbb F$-representable matroids has $n$ nested $k$-separations.

One of the fundamental results in graph minor theory is that for every planar graph $H$, there is a minimum integer $f(H)$ such that graphs with no minor isomorphic to $H$ have treewidth at most $f(H)$. A lower bound for ${f(H)}$ can be…

In this paper we highlight some enumerative results concerning matroids of low rank and prove the tail-ends of various sequences involving the number of matroids on a finite set to be log-convex. We give a recursion for a new, slightly…

Linear rank-width is a linearized variation of rank-width, and it is deeply related to matroid path-width. In this paper, we show that the linear rank-width of every $n$-vertex distance-hereditary graph, equivalently a graph of rank-width…

We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third…

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)…

In this paper, we introduce the problem of Matroid-Constrained Vertex Cover: given a graph with weights on the edges and a matroid imposed on the vertices, our problem is to choose a subset of vertices that is independent in the matroid,…

A minor-closed class of matroids is (strongly) fractal if the number of n-element matroids in the class is dominated by the number of n-element excluded minors. We conjecture that when K is an infinite field, the class of K-representable…

Dumas, Foucaud, Perez and Todinca (2024) recently proved that every graph whose edges can be covered by $k$ shortest paths has pathwidth at most $O(3^k)$. In this paper, we improve this upper bound on the pathwidth to a polynomial one;…

We show that every $H$-minor-free graph that also excludes a $(k \times k)$-grid as a minor has treewidth/branchwidth bounded from above by a function $f(t,k)$ that is linear in $k$ and polynomial in $t := |V(H)|$. Such a result was proven…

The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First…

For every positive integer $k$, we define the $k$-treedepth as the largest graph parameter $\mathrm{td}_k$ satisfying (i) $\mathrm{td}_k(\emptyset)=0$; (ii) $\mathrm{td}_k(G) \leq 1+ \mathrm{td}_k(G-u)$ for every graph $G$ and every vertex…

A matrix is said to have factor width at most $k$ if it can be written as a sum of positive semidefinite matrices that are non-zero only in a single $k \times k$ principal submatrix. We explore the ``factor-width-$k$ rank'' of a matrix,…

A frame template over a field $\mathbb F$ describes the precise way in which a given $\mathbb F$-representable matroid is close to being a frame matroid. Our main result determines the maximum-rank projective or affine geometry that is…

A connectivity function on a finite set $V$ is a symmetric submodular function $f \colon 2^V \to \mathbb{Z}$ with $f(\emptyset)=0$. We prove that finding a branch-decomposition of width at most $k$ for a connectivity function given by an…

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,…

Oriented matroids (often called order types) are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a…

The Theta rank of a finite point configuration $V$ is the maximal degree necessary for a sum-of-squares representation of a non-negative linear function on $V$. This is an important invariant for polynomial optimization that is in general…

In the companion paper [Linear rank-width of distance-hereditary graphs I. A polynomial-time algorithm, Algorithmica 78(1):342--377, 2017], we presented a characterization of the linear rank-width of distance-hereditary graphs, from which…

Let $\mathcal{G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of $\mathcal{G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to $\mathcal{G}.$ We denote by $\mathcal{A}_k…