Computing Treedepth Obstructions

The graph parameter treedepth is minor-monotone; hence, the class of graphs with treedepth at most $k$ is minor-closed. By the Graph Minor Theorem, such a class is characterized by a finite set of forbidden minors. A conjecture of Dvo\v{r}\'ak, Giannopoulou, and Thilikos states that every such forbidden minor has at most $2^k$ vertices. We present an algorithm that, given $n, k \in \mathbb{N}$, computes the set of forbidden minors, forbidden subgraphs, and forbidden induced subgraphs on at most $n$ vertices, for the class of graphs of treedepth at most $k$. Applying this algorithm to $k = 4$ and $n = 16$, we enumerate 1546 forbidden minors, 1718 forbidden subgraphs, and 12204 forbidden induced subgraphs. Assuming the above conjecture holds, these sets constitute the complete obstruction sets for graphs of treedepth at most 4.