User:Isokaedra Aetou/Nowhere dense (graph theory)

In graph theory, a class C of finite graphs is nowhere dense, if it does not contain large subdivided complete graphs. More precisely, C is nowhere dense, if for every integer r≥0 there exists an integer n≥0 such that no graph in C contains the r-subdivision of the complete graph Kn as a subgraph.

Nowhere dense graph classes were introduced by Nešetřil and Ossona de Mendez. The original definition Let C be a class of finite graphs. Let C ∇r denote the class of all r-minors of graphs in C. In particular, C ∇0 is the class of graphs that are subgraphs of a graph in C.

They proved that as r goes to infinity, there are only three possible asymptotic behaviours for the growth of the number of edges of r-minors in terms of their number of vertices: finitely bounded, linear, or quadratic:

$$\lim_{r\to\infty} \limsup_{G\in C\nabla r} \tfrac{\log|E(G)|}{\log|V(G)|}\in\{0,1,2\},$$

where |E(G)| denotes the number of edges of the graph G and |V(G)| denotes the number of vertices of G. They call the class C nowhere dense. if the right-hand side is 1. This is equivalent to the first definition. Moreover, a graph class C is nowhere dense, if and only if for every integer r≥0 there exists a graph Hr such that no graph in C contains Hr as an r-minor.

Examples of nowhere dense graph classes are: trees, planar graphs, graph classes of bounded degree, classes of graphs excluding a fixed minor, graph classes with bounded expansion.