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Obstructions for tree-depth. (English) Zbl 1273.05212
Nešetřil, Jaroslav (ed.) et al., Extended abstracts of the 5th European conference on combinatorics, graph theory and applications, EuroComb’09, Bordeaux, France, September 7–11, 2009. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 34, 249-253 (2009).
Summary: For every \(k \geq 0\), we define \({\mathcal{G}}_{k}\) as the class of graphs with tree-depth at most \(k\), i.e., the class containing every graph \(G\) admitting a valid colouring \(\rho : V(G) \to \{1,\dots,k\}\) 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 class \(\mathsf{obs}({\mathcal{G}}_{k})\) of minor-minimal elements not belonging in \({\mathcal{G}}_{k}\) for every \(k \geq 0\). We give a precise characterization of \({\mathcal{G}}_{k}\), \(k \leq 3\) and prove a structural lemma for creating graphs \(G \in \mathsf{obs}({\mathcal{G}}_{k})\), \(k>0\). As a consequence, we obtain a precise characterization of all acyclic graphs in \(\mathsf{obs}({\mathcal{G}}_{k})\) and we prove that they are exactly \(\frac{1}{2}2^{2^{k-1}-k}\left(1+2^{2^{k-1}-k}\right)\).
For the entire collection see [Zbl 1239.05008].

MSC:
05C83 Graph minors
05C30 Enumeration in graph theory
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