9 problems

Let $\mathcal{C}$ be a sparse hereditary class of graphs with unbounded tree-width. A hereditary graph class is minimal of unbounded tree-width if it has unbounded tree-width, whil…

Tree-width conjecture. The invariant $mu(G)$ is a function only of the tree width of $G$.

Let $K,n,m,r\in\mathbb{N}$. A graph is $(M,A)$-quasi-isometric to a graph of tree-width at most $g$ in the usual coarse sense, and connected sets are at least $K$ apart when every…

Let $K,d,n\in\mathbb{N}$. A graph has a $K$-fat $(n\times n)$-grid minor when it contains the corresponding fat minor model. Then there exists some $g\in\mathbb{N}$ such that every…

Let $n\in\mathbb{N}$. An induced $(n\times n)$-grid minor is the indicated induced-minor model, and $(M,A)$-quasi-isometry and tree-width have their usual meanings. Then there exis…

Let $K,n \in \mathbb{N}$. A graph has a $K$-fat $(n\times n)$-grid minor if it contains the corresponding fat minor model, and graphs are $(M,A)$-quasi-isometric when they satisfy…

Nguyen–Scott–Seymour conjecture. There is a constant $k$ such that if a graph $G$ admits a quasi-isometry to a graph of tree-width at most two, then $G$ admits a quasi-isometry wit…

Radial tree-width conjecture. There is a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that, if a connected graph $G$ does not contain a $c$-quasi-geodesic $(\geqslant ck)$-sub…

Let $\mathcal{C}$ be a class of graphs of twin-width at most $3$, and suppose there exists an integer $t$ such that no graph $G\in\mathcal{C}$ contains $K_{t,t}$ as a subgraph. Spa…