  The tree-depth of $G$ is the smallest value of $k$ for which a labeling of
the vertices of $G$ with elements from $\{1,\dots,k\}$ exists such that any
path joining two vertices with the same label contains a vertex having a higher
label. The graph $G$ is $k$-critical if it has tree-depth $k$ and every proper
minor of $G$ has smaller tree-depth.
  Motivated by a conjecture on the maximum degree of $k$-critical graphs, we
consider the property of 1-uniqueness, wherein any vertex of a critical graph
can be the unique vertex receiving label 1 in an optimal labeling. Contrary to
an earlier conjecture, we construct examples of critical graphs that are not
1-unique and show that 1-unique graphs can have arbitrarily many more edges
than certain critical spanning subgraphs. We also show that $(n-1)$-critical
graphs are 1-unique and use 1-uniqueness to show that the Andr\'{a}sfai graphs
are critical with respect to tree-depth.
