Related papers: Excluding a rectangular grid

Let $G$ be an undirected graph. We say that $G$ contains a ladder of length $k$ if the $2 \times (k+1)$ grid graph is an induced subgraph of $G$ that is only connected to the rest of $G$ via its four cornerpoints. We prove that if all the…

The Gram dimension $\gd(G)$ of a graph $G$ is the smallest integer $k\ge 1$ such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding to edges of $G$, can be…

We show that every graph $G$ excluding an apex-forest $H$ as a minor has layered pathwidth at most $|V(H)|-2$, and that every graph $G$ excluding an apex-linear forest (such as a fan) $H$ as a minor has layered treedepth at most $|V(H)|-2$.…

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…

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…

A graph $G$ is said to be $k$-extendable if every matching of size $k$ in $G$ can be extended to a perfect matching of $G$, where $k$ is a positive integer. We say $G$ is $1$-excludable if for every edge $e$ of $G$, there exists a perfect…

We determine if the width of a graph class ${\cal G}$ changes from unbounded to bounded if we consider only those graphs from ${\cal G}$ whose diameter is bounded. As parameters we consider treedepth, pathwidth, treewidth and clique-width,…

For a tree decomposition $\mathcal{T}$ of a graph $G$, let $\mu(\mathcal{T})$ denote the maximum size of an induced matching in $G$ with the property that some bag of $\mathcal{T}$ contains at least one endpoint of every edge of the…

The GG-width of a class of graphs GG is defined as follows. A graph G has GG-width k if there are k independent sets N1,...,Nk in G such that G can be embedded into a graph H in GG such that for every edge e in H which is not an edge in G,…

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…

We study embeddings of graphs with bounded treewidth or bounded simple treewidth into the undirected graph underlying the directed product of two directed graphs. If the factors have bounded maximum indegrees, then the product graph has…

Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each $k$, there is a finite obstruction set $\mathcal{O}_k$ of graphs such that…

Let $k\geq2$ be an integer. A $k$-tree is a tree with maximum degree at most $k$. In this paper, we give a closure result on spanning $k$-trees of graphs with given minimum degree. Let $\delta\geq1$ be an integer, and $G$ be a connected…

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that…

Robertson and Seymour proved that for every finite tree $H$, there exists $k$ such that every finite graph $G$ with no $H$ minor has path-width at most $k$; and conversely, for every integer $k$, there is a finite tree $H$ such that every…

Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth,…

The isolation number of a graph $G$ (also called the vertex-edge domination number of $G$), denoted by $\iota(G)$, is the size of a smallest subset $D$ of the vertex set $V(G)$ of $G$ such that $G-N[D]$ (the graph obtained by deleting the…

A connected graph has tree-depth at most $k$ if it is a subgraph of the closure of a rooted tree whose height is at most $k$. We give an algorithm which for a given $n$-vertex graph $G$, in time $\mathcal{O}(1.9602^n)$ computes the…

For any integer $k\geq1,$ a graph $G$ has a $k$-factor if it contains a $k$-regular spanning subgraph. In this paper we prove a sufficient condition in terms of the number of $r$-cliques to guarantee the existence of a $k$-factor in a graph…

Let $\mathcal{D}_k$ be the class of graphs for which every minor has minimum degree at most $k$. Then $\mathcal{D}_k$ is closed under taking minors. By the Robertson-Seymour graph minor theorem, $\mathcal{D}_k$ is characterised by a finite…