Related papers: Obstructions for bounded shrub-depth and rank-dept…
In Graph Minors III, Robertson and Seymour write: "It seems that the tree-width of a planar graph and the tree-width of its geometric dual are approximately equal - indeed, we have convinced ourselves that they differ by at most one". They…
A circle graph is an intersection graph of a set of chords of a circle. We describe the unavoidable induced subgraphs of circle graphs with large treewidth. This includes examples that are far from the `usual suspects'. Our results imply…
A variant of the Erd\H{o}s-S\'os conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least $\lfloor 2k/3 \rfloor$ and maximum degree at least $k$ contains a copy of every tree with $k$ edges.…
We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding…
Contraction$^*$-depth is a matroid depth parameter analogous to tree-depth of graphs. We establish the matroid analogue of the classical graph theory result asserting that the tree-depth of a graph $G$ is the minimum height of a rooted…
We give new decomposition theorems for classes of graphs that can be transduced in first-order logic from classes of sparse graphs -- more precisely, from classes of bounded expansion and from nowhere dense classes. In both cases, the…
Tree-width and its linear variant path-width play a central role for the graph minor relation. In particular, Robertson and Seymour (1983) proved that for every tree~$T$, the class of graphs that do not contain $T$ as a minor has bounded…
In the first paper of the Graph Minors series [JCTB '83], Robertson and Seymour proved the Forest Minor theorem: the $H$-minor-free graphs have bounded pathwidth if and only if $H$ is a forest. In recent years, considerable effort has been…
For intractable problems on graphs of bounded treewidth, two graph parameters treedepth and vertex cover number have been used to obtain fine-grained complexity results. Although the studies in this direction are successful, we still need a…
Given two $n$-vertex graphs $G_1$ and $G_2$ of bounded treewidth, is there an $n$-vertex graph $G$ of bounded treewidth having subgraphs isomorphic to $G_1$ and $G_2$? Our main result is a negative answer to this question, in a strong…
We consider classes of arbitrary (finite or infinite) graphs of bounded shrub-depth, specifically the classes $\mathrm{TM}_r(d)$ of arbitrary graphs that have tree models of height $d$ and $r$ labels. We show that the graphs of…
DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid $U_{n,2n}$ or the cycle…
We prove that for all $0\leq t\leq k$ and $d\geq 2k$, every graph $G$ with treewidth at most $k$ has a `large' induced subgraph $H$, where $H$ has treewidth at most $t$ and every vertex in $H$ has degree at most $d$ in $G$. The order of $H$…
Birmele [J. Graph Theory, 2003] proved that every graph with circumference t has treewidth at most t-1. Under the additional assumption of 2-connectivity, such graphs have bounded pathwidth, which is a qualitatively stronger result.…
In this paper and a companion paper, we prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as…
Quasi-isometry is a measure of how similar two graphs are at `large-scale'. Nguyen, Scott, and Seymour [arXiv:2501.09839] and Hickingbotham [arXiv:2501.10840] independently gave a characterisation of graphs quasi-isometric to graphs of…
We prove that the tree-width of graphs in a hereditary class defined by a finite set $F$ of forbidden induced subgraphs is bounded if and only if $F$ includes a complete graph, a complete bipartite graph, a tripod (a forest in which every…
In recent years, there has been significant interest in characterizing the induced subgraph obstructions to bounded treewidth and pathwidth. While this has recently been resolved for pathwidth, the case of treewidth remains open, and prior…
We prove that a hereditary graph class $\mathcal{G}$ defined by finitely many excluded induced subgraphs has bounded tree-$\alpha$ if and only if it is "$(\mathrm{tw},\omega)$-bounded" (that is, for all $t\in \mathbb N$, the class of all…
The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. A reduced graph $G$ is said to be maximal if any reduced…