2019/03/31 by Matt DeVos, O‐joung Kwon, O-joung Kwon +2 · 2 citations
Computer Science · Mathematics · #1-planar graph #Advanced Graph Theory Research #Bounded function #Chordal graph #Combinatorics #Discrete mathematics #Graph #Graph theory and applications #Graphic matroid #Interconnection Networks and Systems #Mathematics #Matroid #Tree (set theory) #Tree-depth #math.CO #msc:05C75
paper · pdf · doi:10.1016/j.ejc.2020.103186
published as European J. Combin., 90(December 2020), 103186 · 36 pages, 2 figures. Final version
openalex publication_date 2020/07/22 · arxiv created 2020/11/04 · arxiv updated 2020/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.