Related papers: Graphs of Linear Growth have Bounded Treewidth

Treedepth, a more restrictive graph width parameter than treewidth and pathwidth, plays a major role in the theory of sparse graph classes. We show that there exists a constant $C$ such that for every positive integers $a,b$ and a graph…

Given a simple graph $G$, its line graph, denoted by $L(G)$, is obtained by representing each edge of $G$ as a vertex, with two vertices in $L(G)$ adjacent whenever the corresponding edges in $G$ share a common endpoint. By applying the…

We prove that every graph of rankwidth at least $72r$ contains an induced subgraph whose minimum balanced cutrank is at least $r$, which implies a vertex subset where every balanced separation has $\mathbb{F}_2$-cutrank at least $r$. This…

A "tree-partition" of a graph $G$ is a partition of $V(G)$ such that identifying the vertices in each part gives a tree. It is known that every graph with treewidth $k$ and maximum degree $\Delta$ has a tree-partition with parts of size…

Product structure theorems are a collection of recent results that have been used to resolve a number of longstanding open problems on planar graphs and related graph classes. One particularly useful version states that every planar graph…

We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more…

We prove that for every graph $G$ with $n$ vertices, the treewidth of $G$ plus the treewidth of the complement of $G$ is at least $n-2$. This bound is tight.

A graph $A$ is "apex" if $A-z$ is planar for some vertex $z\in V(A)$. Eppstein [Algorithmica, 2000] showed that for a minor-closed class $\mathcal{G}$, the graphs in $\mathcal{G}$ with bounded radius have bounded treewidth if and only if…

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;…

For every $r \in \mathbb{N}$, let $\theta_r$ denote the graph with two vertices and $r$ parallel edges. The $\theta_r$-girth of a graph $G$ is the minimum number of edges of a subgraph of $G$ that can be contracted to $\theta_r$. This…

We prove that the (divisorial) gonality of a finite connected graph is lower bounded by its treewidth. We show that equality holds for grid graphs and complete multipartite graphs. We prove that the treewidth lower bound also holds for…

Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour (2006). We investigate the asymptotic behavior of rank-width of a random graph G(n,p). We show that, asymptotically almost surely, (i)…

We consider hereditary classes of graphs equipped with a total order. We provide multiple equivalent characterisations of those classes which have bounded twin-width. In particular, we prove a grid theorem for classes of ordered graphs…

In a reduction sequence of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying $u$ and $v$, each edge incident to exactly one of $u$ and $v$ is coloured red. Bonnet, Kim, Thomass\'e…

Treewidth is an important structural graph parameter that quantifies how closely a graph resembles a tree-like structure. It has applications in many algorithmic and combinatorial problems. In this paper, we study the treewidth of outer…

The girth of a graph $G$ is the length of a shortest cycle of $G$. Jiang (JCT-B, 2001) showed that every graph $G$ with girth at least $2\ell+1$ and minimum degree at least $k/\ell$ contains every tree $T$ with $k$ edges whose maximum…

We consider a class of growing random graphs obtained by creating vertices sequentially one by one: at each step, we choose uniformly the neighbours of the newly created vertex; its degree is a random variable with a fixed but arbitrary…

We prove that every planar graph is contained in $H_1\boxtimes H_2\boxtimes K_2$ for some graphs $H_1$ and $H_2$ both with treewidth 2. This resolves a question of Liu, Norin and Wood [arXiv:2410.20333]. We also show this result is best…

Treewidth is a graph parameter that plays a fundamental role in several structural and algorithmic results. We study the problem of decomposing a given graph $G$ into node-disjoint subgraphs, where each subgraph has sufficiently large…

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…