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

Authors

Michael D. Barrus, University of Rhode Island
John Sinkovic, University of Waterloo

Document Type

Article

Date of Original Version

7-1-2018

Abstract

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.

Publication Title, e.g., Journal

Discrete Mathematics

Volume

341

Issue

7

Citation/Publisher Attribution

Barrus, Michael D., and John Sinkovic. "On 1-uniqueness and dense critical graphs for tree-depth." Discrete Mathematics 341, 7 (2018): 1973-1982. doi: 10.1016/j.disc.2018.03.026.