Combinatorial and Algorithmic Aspects of Monadic Stability

TL;DR

This paper extends combinatorial and algorithmic properties from nowhere dense classes to monadically stable classes of graphs, providing new bounds, regularity lemmas, and kernelization results.

Contribution

It introduces improved bounds on Ramsey numbers, generalizes results on biclique subdivisions, and develops stronger regularity lemmas and polynomial kernels for monadically stable classes.

Findings

Bound on Ramsey numbers $R(s,t)$ improved to $ ext{O}(t^{s-1- ext{delta}})$

Graphs containing subdivided bicliques also contain larger bicliques in monadically stable classes

Stronger regularity lemma and polynomial kernels for independent set and dominating set problems

Abstract

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 $\mathcal{O}(t^{s-1-\delta})$ for some $\delta>0$, improving the bound $R(s,t)\in \mathcal{O}(t^{s-1}/(\log t)^{s-1})$ known for general graphs and the bounds known for $k$-stable graphs when $s\leq k$. - For every monadically stable class $\mathcal{C}$ and every integer $r$, there exists $\delta > 0$ such that every graph $G \in \mathcal{C}$ that contains an $r$-subdivision of the biclique $K_{t,t}$ as a subgraph also contains $K_{t^\delta,t^\delta}$ 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.