Branch-depth: Generalizing tree-depth of graphs

Abstract

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph G=(V,E) and a subset A of E we let λG(A) be the number of vertices incident with an edge in A and an edge in E∖A. For a subset X of V, let ρG(X) be the rank of the adjacency matrix between X and V∖X over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions λG has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions ρG has bounded branch-depth, which we call the rank-depth of graphs.Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by restriction.