Related papers: Grid Minors and Products
An outerstring graph is the intersection graph of curves lying inside a disk with one endpoint on the boundary of the disk. We show that an outerstring graph with $n$ vertices has treewidth $O(\alpha\log n)$, where $\alpha$ denotes the…
We study the existence and uniqueness problem of compact minimal vertical graphs in $\mathbb{H}^n\times\mathbb{R}$, $n\geq 2$, over bounded domains in the slice $\mathbb{H}^n\times\{0\}$, with non-connected boundary having a finite number…
We prove that a connected planar graph with $n$ vertices and $n+\mu$ edges has a vertex separator of size $O( \sqrt{\mu} + 1)$, and this separator can be computed in linear time.
The planar graph product structure theorem of Dujmovi\'{c}, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result…
We establish relations between the bandwidth and the treewidth of bounded degree graphs G, and relate these parameters to the size of a separator of G as well as the size of an expanding subgraph of G. Our results imply that if one of these…
We show that, for every n and every surface $\Sigma$, there is a graph U embeddable on $\Sigma$ with at most cn^2 vertices that contains as minor every graph embeddable on $\Sigma$ with n vertices. The constant c depends polynomially on the…
This paper is concerned with the recognition of approximate graph products with respect to the Cartesian product. Most graphs are prime, although they can have a rich product-like structure. The proposed algorithms are based on a local…
Recently, Bapat and Kurata [\textit{Linear Algebra Appl.}, 562(2019), 135-153] defined the Cartesian product of two square matrices $A$ and $B$ as $A\oslash B=A\otimes \J+\J\otimes B$, where $\J$ is the all one matrix of appropriate order…
We study linkedness of Cartesian product of graphs and prove that the product of an $a$-linked and a $b$-linked graphs is $(a+b-1)$-linked if the graphs are sufficiently large. Further bounds in terms of connectivity are shown. We determine…
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…
This paper studies graphs that have two tree decompositions with the property that every bag from the first decomposition has a bounded-size intersection with every bag from the second decomposition. We show that every graph in each of the…
Chung and Graham [J. London Math. Soc. 1983] claimed to prove that there exists an $n$-vertex graph $G$ with $ \frac{5}{2}n \log_2 n + O(n)$ edges that contains every $n$-vertex tree as a subgraph. Frati, Hoffmann and T\'oth [Combin.…
Several variants of hypergraph products have been introduced as generalizations of the strong and direct products of graphs. Here we show that only some of them are associative. In addition to the Cartesian product, these are the minimal…
The intersection graph of a collection of trapezoids with corner points lying on two parallel lines is called a trapezoid graph. Using binary indexed tree data structure, we improve algorithms for calculating the size and the number of…
We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most…
We study large minors in small-set expanders. More precisely, we consider graphs with $n$ vertices and the property that every set of size at most $\alpha n / t$ expands by a factor of $t$, for some (constant) $\alpha > 0$ and large $t =…
Previous studies has shown that for a weighted undirected graph having $n$ vertices and $m$ edges, a minimal weight spanning tree can be found with $O^*(\sqrt{mn})$ calls to the weight oracle. The present note shows that a given spanning…
We introduce the graph theoretical parameter of edge treewidth. This parameter occurs in a natural way as the tree-like analogue of cutwidth or, alternatively, as an edge-analogue of treewidth. We study the combinatorial properties of…
A ladder is a $2 \times k$ grid graph. When does a graph class $\mathcal{C}$ exclude some ladder as a minor? We show that this is the case if and only if all graphs $G$ in $\mathcal{C}$ admit a proper vertex coloring with a bounded number…
A connected graph is called fragile if it contains an independent vertex cut. In 2002 Chen and Yu proved that every connected graph of order $n$ and size at most $2n-4$ is fragile, and in 2013 Le and Pfender characterized the non-fragile…