Sparse Combinatorial Structures: Classification and Applications

Abstract

We present results of the recent research on sparse graphs and finite structures in the context of contemporary combinatorics, graph theory, model theory and mathematical logic, complexity of algorithms and probability theory. The topics include: complexity of subgraph-and homomorphism-problems; model checking problems for first order formulas in special classes; property testing in sparse classes of structures. All these problems can be studied under the umbrella of classes of structures which are Nowhere Dense and in the context of Nowhere Dense-Somewhere Dense dichotomy. This dichotomy presents the classification of the general classes of structures which proves to be very robust and stable as it can be defined alternatively by most combinatorial extremal invariants as well as by algorithmic and logical terms. We give examples from logic, geometry and extremal graph theory. Finally we characterize the existence of all restricted dualities in terms of limit objects defined on the homomorphism order of graphs.

A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth

“…The notion of classes with bounded expansion were introduced by the authors in [57,58,60], and their properties further studied in [61,62,23,24,63,64,66,67,71,72] and in the monograph [68]. Particularly, classes with bounded expansion include classes excluding a topological minor, like classes with bounded maximum degree, planar graphs, proper minor closed classes, etc.…”

“…The first problem concern existence of modeling FO-limits. Recall that a class C is nowhere dense [65,66,67,69] if, for every integer d there is an integer N such that the d-subdivision of K N is not a subgraph of a graph in C. We have proven, see Theorem 25, that if a monotone class C is such that every FO-convergent sequence of graphs in C has a modeling FO-limit, then C is nowhere dense. It is thus natural to ask whether the converse statement holds.…”

Структурная Разреженность

“…В этой связи авторами в 2006 г. [50], [51] было введено понятие классов с ограниченным подразбиением, которое расширяет понятие классов, исключающих топологический минор (как классов с ограниченными степенями или классов, исключающих минор как планарные графы), а вскоре после этого [56], [57] была сформулирована дихотомия "нигде не плотный / где-то плотный" для классов графов. Несмотря на то что эта дихотомия на первый взгляд выглядит несколько необоснованной, она оказалась очень разумной и выражается несколькими различными способами, эквивалентность которых не очевидна.…”