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FO-Definability of Shrub-Depth

Authors Yijia Chen , Jörg Flum

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Author Details

Yijia Chen
  • Fudan University, Shanghai, China
Jörg Flum
  • Albert-Ludwigs-Universität Freiburg, Germany

Funding

The collaboration of the authors is funded by the Sino-German Center for Research Promotion (GZ 1518). Yijia Chen is also supported by National Natural Science Foundation of China (Project 61872092).

Acknowledgements

We thank Abhisekh Sankaran for discussions concerning problems of the last section. Anonymous reviewers' comments also help to improve the presentation.

Cite AsGet BibTex

Yijia Chen and Jörg Flum. FO-Definability of Shrub-Depth. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.CSL.2020.15

Abstract

Shrub-depth is a graph invariant often considered as an extension of tree-depth to dense graphs. We show that the model-checking problem of monadic second-order logic on a class of graphs of bounded shrub-depth can be decided by AC^0-circuits after a precomputation on the formula. This generalizes a similar result on graphs of bounded tree-depth [Y. Chen and J. Flum, 2018]. At the core of our proof is the definability in first-order logic of tree-models for graphs of bounded shrub-depth.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph theory
Keywords
  • shrub-depth
  • model-checking
  • monadic second-order logic

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