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

Forbidden graphs for tree-depth

For every k ≥ 0, we define Gk as the class of graphs with tree-depth at most k, i.e. the class containing every graph G admitting a valid colouring ρ : V (G) → {1, . . . , k} such that every (x, y)-path between two vertices where ρ(x) = ρ(y) contains a vertex z where ρ(z) > ρ(x). In this paper we study the set of graphs not belonging in Gk that are minimal with respect to the minor/subgraph/ind...

Dense critical and vertex-critical graphs

This paper gives new constructions of k-chromatic critical graphs with high minimum degree and high edge density, and of vertex-critical graphs with high edge density. c © 2002 Elsevier Science B.V. All rights reserved.

On the tree-depth of Random Graphs

The tree–depth is a parameter introduced under several names as a measure of sparsity of a graph. We compute asymptotic values of the tree–depth of random graphs. For dense graphs, p n−1, the tree–depth of a random graph G is a.a.s. td(G) = n − O( √ n/p). Random graphs with p = c/n, have a.a.s. linear tree–depth when c > 1, the tree–depth is Θ(log n) when c = 1 and Θ(log log n) for c < 1. The r...

Shrub-depth: Capturing Height of Dense Graphs

The recent increase of interest in the graph invariant called tree-depth and in its applications in algorithms and logic on graphs led to a natural question: is there an analogously useful “depth” notion also for dense graphs (say; one which is stable under graph complementation)? To this end we introduced, in a 2012 conference paper, the notion of shrub-depth of a graph class which is related ...

Tree-depth of Graphs: Characterisations and Obstructions