Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weakly-sparse and strongly flip-flat classes of graphs are uniformly almost-wide

Fatemeh Ghasemi, Julien Grange, Mamadou Moustapha Kanté, Florent Madelaine

TL;DR

The paper investigates the relationship between two graph-class properties—weak sparsity and strong flip-flatness—and a sparsity-variance notion called uniformly almost-wide. It proves that a hereditary graph class that is both weakly sparse and strongly flip-flat is exactly uniformly almost-wide, using a combinatorial framework built on flips, a key lemma relating flips to small separators, and Ramsey-type arguments, complemented by transduction-closure properties. The work also shows that FO-transductions preserve strong flip-flatness and connects these results to structurally bounded expansion, yielding corollaries that align with known characterisations of nowhere denseness and related sparsity hierarchies. By bridging flip-flatness with uniform wideness, the authors contribute toward a broader conjecture that monadically stable classes correspond to structurally nowhere dense classes, with implications for FO-model checking. The findings extend our understanding of how dense graph classes can be tamed via controlled flips and transductions, and they corroborate related recent results in the literature.

Abstract

In this work we take a step towards characterising strongly flip-flat classes of graphs. Strong flip-flatness appears to be the analogue of uniform almost-wideness in the setting of dense classes of graphs. We prove that strongly flip-flat classes of graphs that are weakly sparse are indeed uniformly almost-wide.

Weakly-sparse and strongly flip-flat classes of graphs are uniformly almost-wide

TL;DR

The paper investigates the relationship between two graph-class properties—weak sparsity and strong flip-flatness—and a sparsity-variance notion called uniformly almost-wide. It proves that a hereditary graph class that is both weakly sparse and strongly flip-flat is exactly uniformly almost-wide, using a combinatorial framework built on flips, a key lemma relating flips to small separators, and Ramsey-type arguments, complemented by transduction-closure properties. The work also shows that FO-transductions preserve strong flip-flatness and connects these results to structurally bounded expansion, yielding corollaries that align with known characterisations of nowhere denseness and related sparsity hierarchies. By bridging flip-flatness with uniform wideness, the authors contribute toward a broader conjecture that monadically stable classes correspond to structurally nowhere dense classes, with implications for FO-model checking. The findings extend our understanding of how dense graph classes can be tamed via controlled flips and transductions, and they corroborate related recent results in the literature.

Abstract

In this work we take a step towards characterising strongly flip-flat classes of graphs. Strong flip-flatness appears to be the analogue of uniform almost-wideness in the setting of dense classes of graphs. We prove that strongly flip-flat classes of graphs that are weakly sparse are indeed uniformly almost-wide.

Paper Structure

This paper contains 4 sections, 7 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result
  4. Conclusion

Key Result

Theorem 1

Let $\mathcal{C}$ be a hereditary class of graphs. Suppose that there is some $k \in \mathbb{N}$ such that, for all $r \in \mathbb{N}$, there is a function $f_r:\mathbb{N} \to \mathbb{N}$ satisfying that, for every $m \in \mathbb{N}$ and every $G \in \mathcal{C}$ of size at least $f_r(m)$, there is Then extension preservation holds over $\mathcal{C}$.

Figures (2)

  • Figure 1: Assuming $|A|> N_r(m)$ is a subset of vertices of $G$, a graph in a uniformly quasi-wide class of graphs $\mathscr{C}$, then there are sets of vertices $S$ (the red crosses), with $|S|\leqslant s_r$, and $B\subseteq A$, with $|B|\geqslant m$, such that $B$ is $r$-independent in $G\setminus S$. The function $N_r$ and the constant $s_r$ both depend on $r$ and the class $\mathscr{C}$. When $s_r$ is independent from $r$ and only depends on the class, we call that class uniformly almost-wide.
  • Figure 2: Assuming $|A|> N_r(m)$ is a subset of vertices of $G$, a graph in a flip-flat class of graphs $\mathscr{C}$, then there is a set of flips $\mathcal{F}$ (the red ellipses), with $|\mathcal{F}|\leqslant s_r$, and a set $B\subseteq A$, with $|B|\geqslant m$, such that $B$ is $r$-independent in $G\oplus \mathcal{F}$. The function $N_r$ and the constant $s_r$ both depend on $r$ and the class $\mathscr{C}$. When $s_r$ is independent from $r$ and only depends on the class, we call that class strongly flip-flat.

Theorems & Definitions (14)

  • Theorem 1: eleftheriadis2024extension
  • Conjecture 2
  • Definition 3
  • Definition 4
  • Definition 5: Flips
  • Corollary 8
  • Definition 9
  • Theorem 10: dreier2023indiscernibles
  • Proposition 11: eleftheriadis2024extension
  • Theorem 12
  • ...and 4 more