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The tree-depth of G is the smallest value of k for which a labeling of the vertices of G with elements from {1,…,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 on n vertices are 1-unique and use 1-uniqueness to show that the Andrásfai graphs are critical with respect to tree-depth.

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Discrete Mathematics