  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 any proper minor of G has smaller tree-depth, and it 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.
  We present several classes of graphs that are both k-critical and 1-unique,
providing examples that satisfy conjectures on critical graphs discussed in
[M.D. Barrus and J. Sinkovic, Uniqueness and minimal obstructions for
tree-depth, submitted].
