Forbidden graphs for tree-depth

For every k ≥ 0 , we define G k as the class of graphs with tree-depth at most k , i.e. the class containing every graph G admitting a valid colouring ρ : V ( G ) → { 1 , … , k } such that every ( x , y ) -path between two vertices where ρ ( x ) = ρ ( y ) contains a vertex z where ρ ( z ) > ρ ( x ) . In this paper, we study the set of graphs not belonging in G k that are minimal with respect to the minor/subgraph/induced subgraph relation (obstructions of G k ). We determine these sets for k ≤ 3 for each relation and prove a structural lemma for creating obstructions from simpler ones. As a consequence, we obtain a precise characterization of all acyclic obstructions of G k and we prove that there are exactly 1 2 2 2 k − 1 − k ( 1 + 2 2 k − 1 − k ) . Finally, we prove that each obstruction of G k has at most 2 2 k − 1 vertices.