On 1-uniqueness and dense critical graphs for tree-depth

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.