Nešetřil J., de Mendez P.O. Sparsity: Graphs, Structures, and Algorithms

Springer, 2012. — 470 p. — (Algorithms and Combinatorics 28). — ISBN: 3642278744.This is the first book devoted to the systematic study of sparse graphs and sparse finite structures. Although the notion of sparsity appears in various contexts and is a typical example of a hard to define notion, the authors devised an unifying classification of general classes of structures. This approach is very robust and it has many remarkable properties. For example the classification is expressible in many different ways involving most extremal combinatorial invariants.
This study of sparse structures found applications in such diverse areas as algorithmic graph theory, complexity of algorithms, property testing, descriptive complexity and mathematical logic (homomorphism preservation,fixed parameter tractability and constraint satisfaction problems). It should be stressed that despite of its generality this approach leads to linear (and nearly linear) algorithms.
Jaroslav Nešetřil is a professor at Charles University, Prague; Patrice Ossona de Mendez is a CNRS researcher et EHESS, Paris.
This book is related to the material presented by the first author at ICM 2010.A Few Problems
Prolegomena
Measuring Sparsity
Classes and Their Classification
Bounded Height Trees and Tree-Depth
Decomposition
Independence
First-Order Constraint Satisfaction Problems, Limits and Homomorphism Dualities
Preservation Theorems
Restricted Homomorphism Dualities
Counting
Back to Classes
Classes with Bounded Expansion – Examples
Some Applications
Property Testing, Hyperfiniteness and Separators
Core Algorithms
Algorithmic Applications
Further Directions
Solutions and Hints for some of the Exercises