8 problems

Induced-minor characterization conjecture. For every integer $t$ there exist integers $c,d$ such that every graph $G$ either contains $K_{t,t}$ or $\boxplus_t$ as an induced minor…

A graph class is fractionally tree-alpha-fragile if it has the fractional $\mathsf{tree}\textnormal{-}\alpha$-fragility property; independence degeneracy is the graph parameter def…

Let $P_t$ denote the $t$-vertex path and $K_{\ell,\ell}$ the complete bipartite graph with … , a graph is … or $K_{,}$. Path–biclique tree-independence conjecture. For every two po…

For a family $\mathcal{F}$ of graphs, a graph $G$ is $\mathcal{F}$-free if no induced subgraph of $G$ is isomorphic to a graph in $\mathcal{F}$. Let $P_r$ be the path with $r$ vert…

Let $\mathcal C$ be a hereditary graph class. A graph class is $(\operatorname{tw},\omega)$-bounded if there is a function $f$ such that $\operatorname{tw}(G)\leq f(\omega(G))$ for…

Gartland's conjecture. For every positive integer $t$, there is an integer $d=d(t)$ such that every $\mathcal{L}_t$-free graph $G$ with no induced subgraph isomorphic to a subdivis…

A graph class is hereditary if it is closed under taking induced subgraphs. The tree-independence number of a graph is the minimum, over its tree-decompositions, of the largest ind…

Let $\ell$ be an integer, let $\phi$ be a CMSO$_2$ formula, and let $G$ be a graph of tree independence number at most $k$, with a weight function $w:V(G)\rightarrow\mathbb{N}$. Th…