Treelike decompositions for transductions of sparse graphs

Dreier, Jan; Gajarský, Jakub; Kiefer, Sandra; Pilipczuk, Michał; Toruńczyk, Szymon

doi:10.48550/arxiv.2201.11082

Cited by 1 publication

(2 citation statements)

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“…Intuitively, a first-order transduction transforms graphs by first nondeterministically adding colors and then reinterpreting the edge relation using a first-order formula φ(x, y), see e.g. [8,13,14,20,24]. This has led to the notion of structurally nowhere dense classes, which are first-order transductions of nowhere dense classes.…”

Section: Introductionmentioning

confidence: 99%

“…Things look better on structurally nowhere dense classes: For every such class C , ϵ > 0 and integer s, it holds that every K s -free n-vertex graph from C contains an independent set of size Ω s,ϵ,C (n 1−ϵ ). This follows from the fact that graphs from structurally nowhere dense classes can be partitioned into O C ,ϵ,s (n ϵ ) parts, each of which induces a cograph [8], and since cographs are perfect, they satisfy R(s, t) = st. (To be more precise, it is proved in [8] that each of the parts has bounded shrubdepth and it is proved in [25] that every graph of bounded shrubdepth can be decomposed into a bounded number of cographs. )…”

Section: Introductionmentioning

confidence: 99%

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Combinatorial and Algorithmic Aspects of Monadic Stability

Dreier¹,

Nikolas²,

Mouawad³

et al. 2022

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Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. Monadically stable classes, originating from model theory, generalize nowhere dense classes and close them under transductions, i.e. transformations defined by colorings and simple first-order interpretations. In this work we aim to extend some combinatorial and algorithmic properties of nowhere dense classes to monadically stable classes of finite graphs. We prove the following results.In monadically stable classes the Ramsey numbers R(s, t) are bounded from above by O(t s−1−δ ) for some δ > 0, improving the bound R(s, t) ∈ O(t s−1 /(log t) s−1 ) known for general graphs and the bounds known for k-stable graphs when s ≤ k. For every monadically stable class C and every integer r, there exists δ > 0 such that every graph G ∈ C that contains an r-subdivision of the biclique Kt,t as a subgraph also contains K t δ ,t δ as a subgraph. This generalizes earlier results for nowhere dense graph classes. We obtain a stronger regularity lemma for monadically stable classes of graphs. Finally, we show that we can compute polynomial kernels for the independent set and dominating set problems in powers of nowhere dense classes. Formerly, only fixed-parameter tractable algorithms were known for these problems on powers of nowhere dense classes.

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Section: Introductionmentioning

confidence: 99%