Transducing paths in graph classes with unbounded shrubdepth

@article{Pilipczuk2022TransducingPI,
title={Transducing paths in graph classes with unbounded shrubdepth},
author={Michal Pilipczuk and Patrice Ossona de Mendez and Sebastian Siebertz},
journal={ArXiv},
year={2022},
volume={abs/2203.16900}
}
• Published 31 March 2022
• Mathematics
• ArXiv
Transductions are a general formalism for expressing transformations of graphs (and more generally, of relational structures) in logic. We prove that a graph class C can be 𝖥𝖮 -transduced from a class of bounded-height trees (that is, has bounded shrubdepth ) if, and only if, from C one cannot 𝖥𝖮 -transduce the class of all paths. This establishes one of the three remaining open questions posed by Blumensath and Courcelle about the 𝖬𝖲𝖮 -transduction quasi-order, even in the stronger form…

References

SHOWING 1-10 OF 26 REFERENCES
Rankwidth meets stability
• Mathematics
SODA
• 2021
Monadic stability and monadic dependence extend classical structural notions for graphs by viewing them in a wider, model-theoretical context by proving that classes with bounded rankwidth excluding some half-graph as a semi-induced subgraph are linearly $\chi$-bounded.
Shrub-depth: Capturing Height of Dense Graphs
• Mathematics
Log. Methods Comput. Sci.
• 2019
An in-depth review of the definition and basic properties of shrub-depth is provided, and its logical aspects which turned out to be most useful are focused on.
Partitions of Graphs into Cographs
• Mathematics
Electron. Notes Discret. Math.
• 2002
Structural Properties of the First-Order Transduction Quasiorder
• Mathematics
CSL
• 2022
It is proved that FO transductions of the class of paths are exactly perturbations of classes with bounded bandwidth, that the local variants of monadic stability and monadic dependence are equivalent to their (standard) non-local versions, and that the classes with pathwidth at most k form a strict hierarchy in the FO transduction quasiorder.
On set systems definable in sparse graph classes, discrepancy, and quantifier elimination
• Mathematics, Computer Science
• 2021
It is shown that the set systems on a ground set U definable on a monotone nowhere dense class of graphs C have hereditary discrepancy at most |U | (for some c < 1/2) and that, on the contrary, for every monotones somewhere dense class C, there is a set system of d-tuples definable in C with discrepancy Ω(|U |1/2).
First-Order Interpretations of Bounded Expansion Classes
• Mathematics, Computer Science
ICALP
• 2018
Classes of graphs with structurally bounded expansion are introduced, defined as first-order transductions of classes of bounded expansion via low treedepth covers (or colorings), replacing treeepth by its dense analogue called shrubdepth.
Obstructions for bounded shrub-depth and rank-depth
• Mathematics
J. Comb. Theory, Ser. B
• 2021
A New Perspective on FO Model Checking of Dense Graph Classes
• Computer Science
2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
• 2016
A structural characterization of graph classes which are FO interpretable in graph classes of bounded degree is given, which allows us to efficiently compute such an interpretation for an input graph.
On the Monadic Second-Order Transduction Hierarchy
• Mathematics
Log. Methods Comput. Sci.
• 2010
The preorder where the authors set C v K if, and only if, there exists a transductionsuch that C � �(K) is studied, is studied to completely describe the resulting hierarchy.
Regularity lemmas for stable graphs
• Mathematics
• 2011
We develop a framework in which Szemeredi's celebrated Regularity Lemma for graphs interacts with core model-theoretic ideas and techniques. Our work relies on a coincidence of ideas from model