7 problems

A bipartite circle graph is a circle graph whose vertices can be divided into two parts such that edges join only vertices in different parts. A graph $H$ is a pivot-minor of $G$ i…

Kanté and Kwon's conjecture. A vertex-minor-closed class of graphs has bounded linear rank-width if and only if it does not contain some tree.

Let $d53d$ be a proper vertex-minor-closed class of graphs. For $k,p displaymath in \mathbb{N}$, a graph is $k$-rank-connected if it has at least $2k$ vertices and satisfies … f…

A graph $H$ is a bipartite circle graph if it is both bipartite and a circle graph. Oum's rank-width conjecture. For every bipartite circle graph $H$, there is an integer $r(H)$ su…

Hliněný–Kwon–Obdržálek–Ordyniak conjecture. The class $\mathcal{C}$ has bounded rank-depth if and only if there exists an integer $t$ such that no graph $G\in\mathcal{C}$ contains…

Let $G$ be a graph, let $\phi(\mathcal{X})$ be a C$_2$MS formula, and let $\alpha$ be an assignment to its free variables. Write $\operatorname{qr}(\phi)$ for the quantifier rank o…

Kanté–Kwon conjecture. For every fixed forest $T$, there is an integer $f(T)$ such that every graph of linear rank-width at least $f(T)$ contains a vertex-minor isomorphic to $T$.