Professors and doctoral students working in the algorithms and graphs area will appreciate this book. Researchers involved in graph structures for computer science and related fields will benefit from the in-depth overview of the book, ranging from contemporary mathematics and graph theory to algorithmic applications. The book provides an impressive collection of published results from many authors, as well as unpublished results from the book’s authors themselves.
The book is very well written and diagrammed to beautifully present the theory supporting the study of sparse and dense objects. It begins with an introduction to the analysis tools for characterization of discrete structures, especially sparse structures. The first of three parts, “Presentation,” provides context for the investigation of combinatorics and its relationship with other mathematical domains. A few problems are delineated in chapter 2, illustrating the scope of the book.
After a brief introduction and the exposition of some typical examples, the second part, “Theory,” is dedicated to the formalization of the main concepts and constructions. It is the largest part of the book, gathering abundant references to enrich readers and support the authors’ theory and research. While chapter 3 introduces all the relevant notions and results that will be used throughout the book, chapter 4 introduces the specific notions used to study density properties, shallow minors, shallow topological minors, and shallow immersions of individual graphs. These notions are applied in chapter 5, leading to the main focus of the book: the nowhere dense and somewhere dense classifications, and the notion of classes with bounded expansion. The importance of classes becomes clear in chapter 5 and, to better understand the end of this chapter, the reader is advised to first read chapter 13. After several classifications are presented in chapter 5, more are given in chapters 7, 8, 11, and 12.
The authors present an interesting characterization that sparse graph properties are intimately related to the properties of trees, especially bounded height trees, and this is proved in chapter 6. The specific relation of low tree-depth coloring with a decomposition theorem is stated in chapter 7, providing the reader with a characterization of the nowhere dense/somewhere dense dichotomy on graphs. This characterization appears many times in different chapters of the book. Chapter 10 presents a connection with model theory permeating first-order logic. This part ends with chapter 13 summarizing the characteristics of nowhere dense classes.
The third part, “Applications,” concerns both theoretical and algorithmic applications of the concepts and results delineated in the second part. It begins by presenting examples of classes with bounded expansions in chapter 14, followed by some applications in chapter 15, including the Burr-Erdős conjecture. Chapter 17 provides core algorithms directly related to the central study of the book, in particular, a fast iterative algorithm to compute a low tree-depth decomposition. Chapter 18 considers algorithmic applications derived from the low tree-depth coloring algorithm.
In summary, the book contains up-to-date research topics laid out in an amazing chain of thoughts. Almost every chapter ends with exercises, aiding professors in advanced graduate courses. The extensive list of references, together with conjectures and open problems, offers professors, students, and researchers alike profound knowledge on the sparsity of graphs, all in one great book.