Laminar set systems consist of non-crossing subsets of a universe with set inclusion essentially corresponding to the descendant relationship of a tree, the so-called laminar tree. Laminar set systems lie at the core of many graph decompositions such as modular decompositions, split decompositions, and bi-join decompositions. We show that from a laminar set system we can obtain the corresponding laminar tree by means of a monadic second order logic (MSO) transduction. This resolves an open question originally asked by Courcelle and is a satisfying resolution as MSO is the natural logic for set systems and is sufficient to define the property "laminar". Using results from Campbell et al. [STACS 2025], we can now obtain transductions for obtaining modular decompositions, co-trees, split decompositions and bi-join decompositions using MSO instead of CMSO. We further gain some insight into the expressive power of counting quantifiers and provide some results towards determining when counting quantifiers can be simulated in MSO in laminar set systems and when they cannot.

We prove that for any circle graph \(H\) with at least one edge and for any positive integer \(k\), there exists an integer \(t = t(k,H)\) so that every graph \(G\) either has a vertex-minor isomorphic to the disjoint union of \(k\) copies of \(H\), or has a \(t\)-perturbation with no vertex-minor isomorphic to \(H\). Using the same techniques, we also prove that for any planar multigraph \(H\), every binary matroid either has a minor isomorphic to the cycle matroid of \(kH\), or is a low-rank perturbation of a binary matroid with no minor isomorphic to the cycle matroid of \(H\).

Let ${\mathbb{F}}$ be a finite field. We prove that there is an MSO-transduction which, given an ${\mathbb{F}}$-representable matroid of path-width k, produces a branch-decomposition of width at most f(k), for some function f. As a corollary, any recognizable property of ${\mathbb{F}}$-representable matroids with bounded path-width is definable in MSO logic, and therefore recognizability is equivalent to MSO-definability on classes of ${\mathbb{F}}$-representable matroids of bounded path-width. This generalizes the result of Bojańczyk, Grohe and Pilipczuk [Logical Methods in Computer Science 17(1), 2021] which asserts the equivalence of the two notions on graphs of bounded linear clique-width.

We show that given a graph G we can CMSO-transduce its modular decomposition, its split decomposition and its bi-join decomposition. This improves results by Courcelle [Logical Methods in Computer Science, 2006] who gave such transductions using order-invariant MSO, a strictly more expressive logic than CMSO. Our methods more generally yield C_{2}MSO-transductions of the canonical decomposition of weakly-partitive set systems and weakly-bipartitive systems of bipartitions.

We consider matroids with the property that every subset of the ground set of size $t$ is contained in both an $\ell$-element circuit and an $\ell$-element cocircuit; we say that such a matroid has the $(t,\ell)$-property. We show that for any positive integer $t$, there is a finite number of matroids with the $(t,\ell)$-property for $\ell<2t$; however, matroids with the $(t,2t)$-property form an infinite family. We say a matroid is a $t$-spike if there is a partition of the ground set into pairs such that the union of any $t$ pairs is a circuit and a cocircuit. Our main result is that if a sufficiently large matroid has the $(t,2t)$-property, then it is a $t$-spike. Finally, we present some properties of $t$-spikes.