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 the restriction.Comment: 34 pages, 2 figure