Hliněný–Kwon–Obdržálek–Ordyniak conjecture on rank-depth

Let $\mathcal{C}$ be a class of graphs. A graph $H$ is a vertex-minor of a graph $G$ if $H$ can be obtained from $G$ by a sequence of local complementations and vertex deletions. The rank-depth of a graph is the parameter discussed above, and a class has bounded rank-depth when its members have rank-depth bounded by a common integer.

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 a path of length $t$ as a vertex-minor.

Rank-depth is monotone under taking vertex-minors, making vertex-minor obstructions natural for classes of bounded rank-depth. The conjecture was verified for graphs of rank-width $1$ by Novotný, but remains open in general.