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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
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