Branch-depth: Generalizing tree-depth of graphs

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...

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...

Tree-depth of Graphs: Characterisations and Obstructions

Tree-depth and Vertex-minors

In a recent paper [6], Kwon and Oum claim that every graph of bounded rank-width is a pivot-minor of a graph of bounded tree-width (while the converse has been known true already before). We study the analogous questions for “depth” parameters of graphs, namely for the tree-depth and related new shrub-depth. We show that shrub-depth is monotone under taking vertex-minors, and that every graph c...

Dynamic-Depth Context Tree Weighting

Reinforcement learning (RL) in partially observable settings is challenging because the agent’s observations are not Markov. Recently proposed methods can learn variable-order Markov models of the underlying process but have steep memory requirements and are sensitive to aliasing between observation histories due to sensor noise. This paper proposes dynamic-depth context tree weighting (D2-CTW)...