Forbidden graphs for tree-depth

  • Authors:

  • Zdenk Dvořák;Archontia C. Giannopoulou;Dimitrios M. Thilikos

  • Affiliations:

  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic;Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece;Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2012

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Abstract

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 @r:V(G)-{1,...,k} such that every (x,y)-path between two vertices where @r(x)=@r(y) contains a vertex z where @r(z)@r(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 122^2^^^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.