Related papers: Linear rankwidth meets stability
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.…
For a graph class $\mathcal G$ and a graph $H$, the four $\mathcal G$-covering numbers of $H$, namely global ${\rm cn}_{g}^{\mathcal{G}}(H)$, union ${\rm cn}_{u}^{\mathcal{G}}(H)$, local ${\rm cn}_{l}^{\mathcal{G}}(H)$, and folded ${\rm…
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…
Many recent works address the question of characterizing induced obstructions to bounded treewidth. In 2022, Lozin and Razgon completely answered this question for graph classes defined by finitely many forbidden induced subgraphs. Their…
Motivated by algorithmic applications, Kun, O'Brien, Pilipczuk, and Sullivan introduced the parameter linear chromatic number as a relaxation of treedepth and proved that the two parameters are polynomially related. They conjectured that…
Ramsey's Theorem states that a graph $G$ has bounded order if and only if $G$ contains no complete graph $K_n$ or empty graph $E_n$ as its induced subgraph. The Gy\'arf\'as-Sumner conjecture says that a graph $G$ has bounded chromatic…
Each hereditary property can be characterized by its set of minimal obstructions; these sets are often unknown, or known but infinite. By allowing extra structure it is sometimes possible to describe such properties by a finite set of…
The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that for a given $k$-uniform hypergraph (or $k$-graph) $H$, if $n$ is sufficiently large then any colouring of the edges of the complete $k$-graph $K^{(k)}_n$ gives rise…
It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show…
For a given class $\mathcal{C}$ of graphs and given integers $m \leq n$, let $f_\mathcal{C}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in any graph belonging to $\mathcal{C}$ have a (possibly partial) rainbow…
We introduce, for every surface {\Sigma}, a two-way connection between FO transductions (first-order logical transformations) of the graphs embeddable in {\Sigma} and a certain variant of fan-crossing drawings of graphs in {\Sigma}. If the…
We prove that for every positive integer k, there exists an MSO_1-transduction that given a graph of linear cliquewidth at most k outputs, nondeterministically, some cliquewidth decomposition of the graph of width bounded by a function of…
Structural graph parameters, such as treewidth, pathwidth, and clique-width, are a central topic of study in parameterized complexity. A main aim of research in this area is to understand the "price of generality" of these widths: as we…
A graph $\mathcal{H}=(W,E_\mathcal{H})$ is said to have {\em bandwidth} at most $b$ if there exists a labeling of $W$ as $w_1,w_2,\dots,w_n$ such that $|i-j|\leq b$ for every edge $w_iw_j\in E_\mathcal{H}$. We say that $\mathcal{H}$ is a…
We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree {\sl rainbow} trees in various host graphs, having their edges coloured independently and uniformly at random, using a predetermined palette.…
Graph classes of bounded tree rank were introduced recently in the context of the model checking problem for first-order logic of graphs. These graph classes are a common generalization of graph classes of bounded degree and bounded…
We continue the study of $(\mathrm{tw},\omega)$-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of understanding the extent to which this…
The twin-width of a graph $G$ is the minimum integer $d$ such that $G$ has a $d$-contraction sequence, that is, a sequence of $|V(G)|-1$ iterated vertex identifications for which the overall maximum number of red edges incident to a single…
Unlike minors, the induced subgraph obstructions to bounded treewidth come in a large variety, including, for every $t\geq 1$, the $t$-basic obstructions: the graphs $K_{t+1}$ and $K_{t,t}$, along with the subdivisions of the $t$-by-$t$…
We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper, one of us proved that…