Dvořák–Giannopoulou–Thilikos conjecture on the order of critical graphs

Let $G$ be a $k$-critical graph, meaning that $\frac{}{}G$ has tree-depth $k$ and every proper minor of $G$ has tree-depth less than $k$.

Dvořák–Giannopoulou–Thilikos conjecture. $G$ has at most

$$2^{k-1}$$

vertices.

This conjecture concerns the maximum order of graphs that are minor-critical for tree-depth. It is attributed in the source to Dvořák, Giannopoulou, and Thilikos and remains open in full generality.