Obstructions 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 class obs ( G k ) of minor-minimal elements not belonging in G k for every k ⩾ 0 . We give a precise characterization of G k , k ⩽ 3 and prove a structural lemma for creating graphs G ∈ obs ( G k ) , k > 0 . As a consequence, we obtain a precise characterization of all acyclic graphs in obs ( G k ) and we prove that they are exactly 1 2 2 2 k − 1 − k ( 1 + 2 2 k − 1 − k ) .