Uniqueness and minimal obstructions for tree-depth

A k -ranking of a graph G is a labeling of the vertices of G with values from { 1 , ? , k } such that any path joining two vertices with the same label contains a vertex having a higher label. The tree-depth of G is the smallest value of k for which a k -ranking of G exists. The graph G is k -critical if it has tree-depth k and every proper minor of G has smaller tree-depth.We establish partial results in support of two conjectures about the order and maximum degree of k -critical graphs. As part of these results, we define a graph G to be 1-unique if for every vertex v in G , there exists an optimal ranking of G in which v is the unique vertex with label 1. We show that several classes of k -critical graphs are 1-unique, and we conjecture that the property holds for all k -critical graphs. Generalizing a previously known construction for trees, we exhibit an inductive construction that uses 1-unique k -critical graphs to generate large classes of critical graphs having a given tree-depth.