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Nowhere dense graph classes, stability, and the independence property

Summary: Nowhere dense graph classes,
stability, and the independence property
Hans Adler and Isolde Adler
17th November 2010
A class of graphs is nowhere dense if for every integer r there is a finite upper bound
on the size of cliques that occur as (topological) r-minors. We observe that this tameness
notion from algorithmic graph theory is essentially the earlier stability theoretic notion
of superflatness. For subgraph-closed classes of graphs we prove equivalence to stability
and to not having the independence property.
1 Introduction
Recently, Nesetril and Ossona de Mendez [11, 12] introduced nowhere dense classes of
finite graphs, a generalisation of many natural and important classes such as graphs of
bounded degree, planar graphs, graphs excluding a fixed minor and graphs of bounded
expansion. These graph classes play an important role in algorithmic graph theory, as
many computational problems that are hard in general become tractable when restricted
to such classes. All these graph classes are nowhere dense. Dawar and Kreutzer [4]
gave efficient algorithms for domination problems on classes of nowhere dense graphs.
Moreover, nowhere dense classes were studied in the area of finite model theory [2, 3] under
the guise of (uniformly) quasi-wide classes and again turn out to be well-behaved.1


Source: Adler, Isolde - Institut für Informatik, Johann Wolfgang Goethe-Universität Frankfurt am Main


Collections: Computer Technologies and Information Sciences