Document Type
Article
Date of Original Version
2015
Embargo Date
2-6-2018
Abstract
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 every proper minor of G has smaller tree-depth.
We establish partial results in support of two conjectures about the order and maximum degree of k-critical graphs. As part of these results, we define a graph G to be 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 show that several classes of k-critical graphs are 1-unique, and we conjecture that the property holds for all k-critical graphs. Generalizing a previously known construction for trees, we exhibit an inductive construction that uses 1-unique k-critical graphs to generate large classes of critical graphs having a given tree-depth.
Citation/Publisher Attribution
Barrus, M. D., & Sinkovic, J. (2016). Uniqueness and Minimal Obstructions for Tree-Depth. Discrete Mathematics, 339(2), 606-613.
Available at: http://dx.doi.org/10.1016/j.disc.2015.09.027
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.