Related papers: Grid Minors and Products

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…

In an EPG-representation of a graph $G$ each vertex is represented by a path in the rectangular grid, and $(v,w)$ is an edge in $G$ if and only if the paths representing $v$ an $w$ share a grid-edge. Requiring paths representing edges to be…

Let Q_n be the graph of n times n times n cube with all non-decreasing diagonals (including the facial ones) in its constituent unit cubes. Suppose that a subset S of V(Q_n) separates the left side of the cube from the right side. We show…

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…

A graph $G$ contains a graph $H$ as an induced minor if $H$ can be obtained from $G$ after vertex deletions and edge contractions. We show that for every $k$-vertex planar graph $H$, every graph $G$ excluding $H$ as an induced minor and…

Let $s(n)$ be the minimum number of edges in a graph that contains every $n$-vertex tree as a subgraph. Chung and Graham [J. London Math. Soc. 1983] claim to prove that $s(n)\leqslant O(n\log n)$. We point out a mistake in their proof. The…

Let $\mathbb{G}_{n,\gamma}$ be the set of simple and connected graphs on $n$ vertices and with domination number $\gamma$. The graph with minimum spectral radius among $\mathbb{G}_{n,\gamma}$ is called the minimizer graph. In this paper, we…

We study a variety of natural constructions from topological combinatorics, including matching complexes as well as other graph complexes, from the perspective of the graph minor category of \parencite{MiProRa}. We prove that these…

Given any $n \geq 2$, we show that if $\Omega \subsetneq \mathbb{R}^n$ is an open convex domain (e.g. a half-space), and $u : \Omega \to \mathbb{R}$ is a solution to the minimal surface equation which agrees with a linear function on…

A Norton algebra is an eigenspace of a distance regular graph endowed with a commutative nonassociative product called the Norton product, which is defined as the projection of the entrywise product onto this eigenspace. The Norton algebras…

Robertson and Seymour proved that the family of all graphs containing a fixed graph $H$ as a minor has the Erd\H{o}s-P\'osa property if and only if $H$ is planar. We show that this is no longer true for the edge version of the…

We prove that, for each circle graph $H$, every graph with sufficiently large rank-width contains a vertex-minor isomorphic to $H$.

We provide a general framework on the coefficients of the graph polynomials of graphs which are Cartesian products. As a corollary, we prove that if $G=(V,E)$ is a graph with degrees of vertices $2d(v), v\in V$, and the graph polynomial…

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].…

Given a 2-crossing minimal chart $\Gamma$, a minimal chart with two crossings, set $\alpha=\min\{~i~|~$there exists an edge of label $i$ containing a white vertex$\}$, and $\beta=\max\{~i~|~$there exists an edge of label $i$ containing a…

Given an undirected $n$-vertex graph $G(V,E)$ and an integer $k$, let $T_k(G)$ denote the random vertex induced subgraph of $G$ generated by ordering $V$ according to a random permutation $\pi$ and including in $T_k(G)$ those vertices with…

The central theorem of topological graph theory states that the graph minor relation is a well-quasi-order on graphs. It has far-reaching consequences, in particular in the study of graph structures and the design of (parameterized)…

Recently, Kawarabayashi and Thorup presented the first deterministic edge-connectivity recognition algorithm in near-linear time. A crucial step in their algorithm uses the existence of vertex subsets of a simple graph $G$ on $n$ vertices…

We show that we can assume graphs that do not have the edge-Erd\H{o}s-P\'{o}sa property to be connected. Then we strengthen this result to $2$-connectivity under the additional assumptions of a minor-closed property and a generic…

For graphs $G,H$ it is possible to add $(|V(G)|-\gamma(G))(|V(H)|-\gamma(H))$ edges to the Cartesian product $G\mathbin{\square}H$ such that a minimal dominating set $D$ of size $\gamma(G)\gamma(H)$ emerges. We hypothesize that $D$ is also…