Minimal obstructions for tree-depth: A non-1-unique example

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. As defined in [M. D. Barrus, J. Sinkovic, Uniqueness and minimal obstructions for tree-depth, Discrete Math 339 (2) (2015) 606-613], a graph G is 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. In the above paper and [M. D. Barrus, J. Sinkovic, Classes of critical graphs for tree-depth, arXiv:1502.0577] the authors showed that several classes of critical graphs are 1-unique and asked whether all critical graphs have this property. We answer in the negative by demonstrating an infinite family of graphs which are critical but not 1-unique.