Pathwidth vs cocircumference

The {\em circumference} of a graph $G$ with at least one cycle is the length of a longest cycle in $G$. A classic result of Birmel\'e (2003) states that the treewidth of $G$ is at most its circumference minus $1$. In case $G$ is $2$-connected, this upper bound also holds for the pathwidth of $G$; in fact, even the treedepth of $G$ is upper bounded by its circumference (Bria\'nski, Joret, Majewski, Micek, Seweryn, Sharma; 2023). In this paper, we study whether similar bounds hold when replacing the circumference of $G$ by its {\em cocircumference}, defined as the largest size of a {\em bond} in $G$, an inclusion-wise minimal set of edges $F$ such that $G-F$ has more components than $G$. In matroidal terms, the cocircumference of $G$ is the circumference of the bond matroid of $G$. Our first result is the following `dual' version of Birmel\'e's theorem: The treewidth of a graph $G$ is at most its cocircumference. Our second and main result is an upper bound of $3k-2$ on the pathwidth of a $2$-connected graph $G$ with cocircumference $k$. Contrary to circumference, no such bound holds for the treedepth of $G$. Our two upper bounds are best possible up to a constant factor.Comment: v2: revised following the referees' comment

Characterization of matrices with bounded graver bases and depth parameters and applications to integer programming

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the \u1d4c1₁-norm of the Graver basis is bounded by a function of the maximum \u1d4c1₁-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the \u1d4c1₁-norm of the Graver basis of the constraint matrix, when parameterized by the \u1d4c1₁-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix

On infinite antichains of matroids

Robertson and Seymour have shown that there is no infinite set of graphs in which no member is a minor of another. By contrast, it is well known that the class of all matroids does contains such infinite antichains. However, for many classes of matroids, even the class of binary matroids, it is not known whether or not the class contains an infinite antichain. In this paper, we examine a class of matroids of relatively simple structure: Ma, b, c consists of those matroids for which the deletion of some set of at most a elements and the contraction of some set of at most b elements results in a matroid in which every component has at most c elements. We determine precisely when Ma, b, c contains an infinite antichain. We also show that, among the matroids representable over a finite fixed field, there is no infinite antichain in a fixed Ma, b, c; nor is there an infinite antichain when the circuit size is bounded. © 1995 by Academic Press, Inc