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arXiv · 1310.1116

Uniqueness and minimal obstructions for tree-depth

Michael D. Barrus·John Sinkovic
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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.

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Michael D. Barrus, John Sinkovic. 2013-10-03. Uniqueness and minimal obstructions for tree-depth. https://doi.org/10.1016/j.disc.2015.09.027

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