Branch-depth: Generalizing tree-depth of graphs. (December 2020) - DeVos, Matt; Kwon, O-joung; Oum, Sang-il
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 λ G ( 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 ρ G ( 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 λ G has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions ρ G 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.