Nowhere dense graph classes and algorithmic applications. A tutorial at Highlights of Logic, Games and Automata 2019

TL;DR

This paper introduces the concept of nowhere dense graph classes, a broad framework for sparse graphs, and discusses their algorithmic applications, especially in fixed-parameter tractable model-checking for first-order logic.

Contribution

It provides a comprehensive tutorial on nowhere dense graph classes, highlighting their properties, characterizations, and significance in algorithmic graph theory.

Findings

Nowhere dense classes generalize many sparse graph classes.

Model-checking for first-order logic is fixed-parameter tractable on these classes.

The tutorial connects theoretical properties with practical algorithmic applications.

Abstract

The notion of nowhere dense graph classes was introduced by Ne\v{s}et\v{r}il and Ossona de Mendez and provides a robust concept of uniform sparseness of graph classes. Nowhere dense classes generalize many familiar classes of sparse graphs such as classes that exclude a fixed graph as a minor or topological minor. They admit several seemingly unrelated natural characterizations that lead to strong algorithmic applications. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over these classes. These notes, prepared for a tutorial at Highlights of Logic, Games and Automata 2019, are a brief introduction to the theory of nowhere denseness, driven by algorithmic applications.