Obstructions for bounded shrub-depth and rank-depth

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Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth, which was conjectured by Hlineny et al. (2016) [11]. Namely, we prove that a graph has large rank-depth if and only if it has a vertex-minor isomorphic to a long path. This implies that for every integer t, the class of graphs with no vertex-minor isomorphic to the path on t vertices has bounded shrub-depth. (C) 2021 Elsevier Inc. All rights reserved.
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
Issue Date
2021-07
Language
English
Article Type
Article
Citation

JOURNAL OF COMBINATORIAL THEORY SERIES B, v.149, pp.76 - 91

ISSN
0095-8956
DOI
10.1016/j.jctb.2021.01.005
URI
http://hdl.handle.net/10203/285298
Appears in Collection
MA-Journal Papers(저널논문)
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