Branch-depth: Generalizing tree-depth of graphsopen access
- Authors
- DeVos,Matt; Kwon, O jung; Oum, Sang-il
- Issue Date
- Dec-2020
- Publisher
- ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
- Citation
- EUROPEAN JOURNAL OF COMBINATORICS, v.90, pp.1 - 23
- Indexed
- SCIE
SCOPUS
- Journal Title
- EUROPEAN JOURNAL OF COMBINATORICS
- Volume
- 90
- Start Page
- 1
- End Page
- 23
- URI
- https://scholarworks.bwise.kr/hanyang/handle/2021.sw.hanyang/189493
- DOI
- 10.1016/j.ejc.2020.103186
- ISSN
- 0195-6698
- 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 lambda(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 rho(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 lambda(G) has bounded branch depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions rho(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.
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Collections - 서울 자연과학대학 > 서울 수학과 > 1. Journal Articles
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