Related papers: Treewidth is Polynomial in Maximum Degree on Weakl…

A graph $H$ is an \emph{induced minor} of a graph $G$ if $H$ can be obtained from $G$ by a sequence of edge contractions and vertex deletions. Otherwise, $G$ is \emph{$H$-induced minor-free}. In this paper, we provide a different proof of…

The product dimension of a graph G is defined as the minimum natural number l such that G is an induced subgraph of a direct product of l complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and…

We show that the eccentricities, diameter, radius, and Wiener index of an undirected $n$-vertex graph with nonnegative edge lengths can be computed in time $O(n\cdot \binom{k+\lceil\log n\rceil}{k} \cdot 2^k k^2 \log n)$, where $k$ is the…

We prove that every sufficiently big 6-connected graph of bounded tree-width either has a K_6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently…

We investigate a structural generalisation of treewidth we call $\mathcal{A}$-blind-treewidth where $\mathcal{A}$ denotes an annotated graph class. This width parameter is defined by evaluating only the size of those bags $B$ of…

A resolving set $S$ of a graph $G$ is a subset of its vertices such that no two vertices of $G$ have the same distance vector to $S$. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a…

We give a short proof that for every apex-forest $X$ on at least two vertices, graphs excluding $X$ as a minor have layered pathwidth at most $2|V(X)|-3$. This improves upon a result by Dujmovi\'c, Eppstein, Joret, Morin, and Wood (SIDMA,…

In 2019, Dvo\v{r}\'{a}k asked whether every connected graph $G$ has a tree decomposition $(T, \mathcal{B})$ so that $T$ is a subgraph of $G$ and the width of $(T, \mathcal{B})$ is bounded by a function of the treewidth of $G$. We prove that…

We prove that the tree-width of graphs in a hereditary class defined by a finite set $F$ of forbidden induced subgraphs is bounded if and only if $F$ includes a complete graph, a complete bipartite graph, a tripod (a forest in which every…

We introduce a new parameter, called stretch-width, that we show sits strictly between clique-width and twin-width. Unlike the reduced parameters [BKW '22], planar graphs and polynomial subdivisions do not have bounded stretch-width. This…

The linear arboricity of a graph $G$, denoted by $\text{la}(G)$, is the minimum number of edge-disjoint linear forests (i.e. forests in which every connected component is a path) in $G$ whose union covers all the edges of $G$. A famous…

We prove that if every subgraph of a graph $G$ has a balanced separation of order at most $a$ then $G$ has treewidth at most $15a$. This establishes a linear dependence between the treewidth and the separation number.

Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work…

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.…

A set $S\subseteq V$ of vertices of a graph $G$ is a $c$-clustered set if it induces a subgraph with components of order at most $c$ each, and $\alpha_c(G)$ denotes the size of a largest $c$-clustered set. For any graph $G$ on $n$ vertices…

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…

A $(\beta,\delta,\Delta)$-padded decomposition of an edge-weighted graph $G = (V,E,w)$ is a stochastic decomposition into clusters of diameter at most $\Delta$ such that for every vertex $v\in V$, the probability that…

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…

This paper explores the structure of graphs defined by an excluded minor or an excluded odd minor through the lens of graph products and tree-decompositions. We prove that every graph excluding a fixed odd minor is contained in the strong…

A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree $k$ or less. We prove that every connected planar graph with average degree $d \ge 2$ has a 4-degenerate induced…