Related papers: Improved Bounds for the Excluded Grid Theorem
We study the Excluded Grid Theorem, a fundamental structural result in graph theory, that was proved by Robertson and Seymour in their seminal work on graph minors. The theorem states that there is a function $f: \mathbb{Z}^+ \to…
One of the key results in Robertson and Seymour's seminal work on graph minors is the Grid-Minor Theorem (also called the Excluded Grid Theorem). The theorem states that for every grid $H$, every graph whose treewidth is large enough…
As part of the graph minor project, Robertson and Seymour showed in 1990 that the class of graphs that can be embedded in a given surface can be characterized by a finite set of minimal excluded minors. However, their proof, because…
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…
Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $\ell\times\ell$ grid minor is exponential in $\ell$. It is unknown…
The celebrated grid exclusion theorem states that for every $h$-vertex planar graph $H$, there is a constant $c_{h}$ such that if a graph $G$ does not contain $H$ as a minor then $G$ has treewidth at most $c_{h}$. We are looking for…
As part of their graph minor project, Robertson and Seymour showed in 1990 that the class of graphs that can be embedded in a given surface can be characterized by a finite set of minimal excluded minors. However, their proof, because…
We prove that for every planar graph $X$ of treedepth $h$, there exists a positive integer $c$ such that for every $X$-minor-free graph $G$, there exists a graph $H$ of treewidth at most $f(h)$ such that $G$ is isomorphic to a subgraph of…
For their famous algorithm for the disjoint paths problem, Robertson and Seymour proved that there is a function $f$ such that if the tree-width of a graph $G$ with $k$ pairs of terminals is at least $f(k)$, then $G$ contains a…
We prove that for every tree $T$ of radius $h$, there is an integer $c$ such that every $T$-minor-free graph is contained in $H\boxtimes K_c$ for some graph $H$ with pathwidth at most $2h-1$. This is a qualitative strengthening of the…
Square grids play a pivotal role in Robertson and Seymour's work on graph minors as planar obstructions to small treewidth. We introduce a three-sided bramble in a plane graph called a net, which generalizes the standard bramble of crosses…
Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger…
The Grid Minor Theorem states that for every planar graph $H$, there exists a smallest integer $f(H)$ such that every graph with tree-width at least $f(H)$ contains $H$ as a minor. The only known lower bounds on $f(H)$ beyond the trivial…
We show that every $H$-minor-free graph that also excludes a $(k \times k)$-grid as a minor has treewidth/branchwidth bounded from above by a function $f(t,k)$ that is linear in $k$ and polynomial in $t := |V(H)|$. Such a result was proven…
A 1993 result of Alon and F\"uredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to…
The Flat Wall Theorem of Robertson and Seymour states that there is some function $f$, such that for all integers $w,t>1$, every graph $G$ containing a wall of size $f(w,t)$, must contain either (i) a $K_t$-minor; or (ii) a small subset…
A graph has tree-width at most $k$ if it can be obtained from a set of graphs each with at most $k+1$ vertices by a sequence of clique sums. We refine this definition by, for each non-negative integer $\theta$, defining the…
By the Grid Minor Theorem of Robertson and Seymour, every graph of sufficiently large tree-width contains a large grid as a minor. Tree-width may therefore be regarded as a measure of 'grid-likeness' of a graph. The grid contains a long…
The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, $K_4$)-free graphs was constructed, that still has unbounded tree-width [Sintiari and Trotignon, 2019].…
The grid theorem, originally proved by Robertson and Seymour in Graph Minors V in 1986, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance…