On nowhere dense and somewhere dense - a graph trichotomy
نویسندگان
چکیده
Motivated by structural decompositions and other asymptotic properties of graphs we define the notions in the title and show the wide spectrum of examples of nowhere dense graphs. This has several algorithmic consequences and it generalizes and improves earlier results.
منابع مشابه
On nowhere dense graphs
A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special cl...
متن کاملNeighborhood Complexity and Kernelization for Nowhere Dense Classes of Graphs
We prove that whenever G is a graph from a nowhere dense graph class C, and A is a subset of vertices of G, then the number of subsets of A that are realized as intersections of A with r-neighborhoods of vertices of G is at most f(r, ε) · |A|, where r is any positive integer, ε is any positive real, and f is a function that depends only on the class C. This yields a characterization of nowhere ...
متن کاملReconfiguration on nowhere dense graph classes
In the token jumping problem for a vertex subset problem Q on graphs we are given a graph G and two feasible solutions Ss, St ⊆ V (G) of Q with |Ss| = |St|, and imagine that a token is placed on each vertex of Ss. The problem is to determine whether there exists a sequence S1, . . . , Sn of feasible solutions, where S1 = Ss, Sn = St and each Si+1 results from Si, 1 ≤ i < n, by moving exactly on...
متن کاملSparse Combinatorial Structures: Classification and Applications
We present results of the recent research on sparse graphs and finite structures in the context of of contemporary combinatorics, graph theory, model theory and mathematical logic, complexity of algorithms and probability theory. The topics include: complexity of subgraphand homomorphismproblems; model checking problems for first order formulas in special classes; property testing in sparse cla...
متن کاملCharacterisations of Nowhere Dense Graphs
Nowhere dense classes of graphs were introduced by Nešetřil and Ossona de Mendez as a model for “sparsity” in graphs. It turns out that nowhere dense classes of graphs can be characterised in many different ways and have been shown to be equivalent to other concepts studied in areas such as (finite) model theory. Therefore, the concept of nowhere density seems to capture a natural property of g...
متن کامل