Branch-depth: Generalizing tree-depth of graphs
- Title
- Branch-depth: Generalizing tree-depth of graphs
- Author
- 권오정
- Issue Date
- 2020-12
- Publisher
- ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
- Citation
- EUROPEAN JOURNAL OF COMBINATORICS, v. 90, article no. 103186
- 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.
- URI
- https://www.sciencedirect.com/science/article/pii/S0195669820301074?via%3Dihubhttps://repository.hanyang.ac.kr/handle/20.500.11754/173188
- ISSN
- 0195-6698; 1095-9971
- DOI
- 10.1016/j.ejc.2020.103186
- Appears in Collections:
- COLLEGE OF NATURAL SCIENCES[S](자연과학대학) > MATHEMATICS(수학과) > Articles
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