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Forbidden graphs for tree-depth. (English) Zbl 1239.05062
Summary: For every $$k\geq 0$$, 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 $$\rho :V(G)\to {1,\ldots ,k}$$ such that every $$(x,y)$$-path between two vertices where $$\rho (x)=\rho (y)$$ contains a vertex $$z$$ where $$\rho (z)>\rho (x)$$.
In this paper, we study the set of graphs not belonging in $$\mathcal G_{k}$$) that are minimal with respect to the minor/subgraph/induced subgraph relation (obstructions of $$\mathcal G_{k}$$). We determine these sets for $$k\leq 3$$ 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 $$\mathcal G_{k}$$) and we prove that there are exactly $$\frac{1}{2}2^{2^{k - 1}-k}(1+2^{2^{k - 1}-k})$$. Finally, we prove that each obstruction of $$G_{k}$$ has at most $$2^{2^{k - 1}}$$ vertices.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C05 Trees
##### Keywords:
acyclic obstructions of a graph
Full Text:
##### References:
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