31,616 research outputs found
Deciding first-order properties of nowhere dense graphs
Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez,
form a large variety of classes of "sparse graphs" including the class of
planar graphs, actually all classes with excluded minors, and also bounded
degree graphs and graph classes of bounded expansion.
We show that deciding properties of graphs definable in first-order logic is
fixed-parameter tractable on nowhere dense graph classes. At least for graph
classes closed under taking subgraphs, this result is optimal: it was known
before that for all classes C of graphs closed under taking subgraphs, if
deciding first-order properties of graphs in C is fixed-parameter tractable,
then C must be nowhere dense (under a reasonable complexity theoretic
assumption).
As a by-product, we give an algorithmic construction of sparse neighbourhood
covers for nowhere dense graphs. This extends and improves previous
constructions of neighbourhood covers for graph classes with excluded minors.
At the same time, our construction is considerably simpler than those. Our
proofs are based on a new game-theoretic characterisation of nowhere dense
graphs that allows for a recursive version of locality-based algorithms on
these classes. On the logical side, we prove a "rank-preserving" version of
Gaifman's locality theorem.Comment: 30 page
On First-Order Definable Colorings
We address the problem of characterizing -coloring problems that are
first-order definable on a fixed class of relational structures. In this
context, we give several characterizations of a homomorphism dualities arising
in a class of structure
On the number of types in sparse graphs
We prove that for every class of graphs which is nowhere dense,
as defined by Nesetril and Ossona de Mendez, and for every first order formula
, whenever one draws a graph and a
subset of its nodes , the number of subsets of which are of
the form
for some valuation of in is bounded by
, for every . This provides
optimal bounds on the VC-density of first-order definable set systems in
nowhere dense graph classes.
We also give two new proofs of upper bounds on quantities in nowhere dense
classes which are relevant for their logical treatment. Firstly, we provide a
new proof of the fact that nowhere dense classes are uniformly quasi-wide,
implying explicit, polynomial upper bounds on the functions relating the two
notions. Secondly, we give a new combinatorial proof of the result of Adler and
Adler stating that every nowhere dense class of graphs is stable. In contrast
to the previous proofs of the above results, our proofs are completely
finitistic and constructive, and yield explicit and computable upper bounds on
quantities related to uniform quasi-wideness (margins) and stability (ladder
indices)
First-order limits, an analytical perspective
In this paper we present a novel approach to graph (and structural) limits
based on model theory and analysis. The role of Stone and Gelfand dualities is
displayed prominently and leads to a general theory, which we believe is
naturally emerging. This approach covers all the particular examples of
structural convergence and it put the whole in new context. As an application,
it leads to new intermediate examples of structural convergence and to a "grand
conjecture" dealing with sparse graphs. We survey the recent developments
Nowhere dense graph classes, stability, and the independence property
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.Comment: 9 page
Testing first-order properties for subclasses of sparse graphs
We present a linear-time algorithm for deciding first-order (FO) properties
in classes of graphs with bounded expansion, a notion recently introduced by
Nesetril and Ossona de Mendez. This generalizes several results from the
literature, because many natural classes of graphs have bounded expansion:
graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs
of bounded degree, graphs with no subgraph isomorphic to a subdivision of a
fixed graph, and graphs that can be drawn in a fixed surface in such a way that
each edge crosses at most a constant number of other edges. We deduce that
there is an almost linear-time algorithm for deciding FO properties in classes
of graphs with locally bounded expansion.
More generally, we design a dynamic data structure for graphs belonging to a
fixed class of graphs of bounded expansion. After a linear-time initialization
the data structure allows us to test an FO property in constant time, and the
data structure can be updated in constant time after addition/deletion of an
edge, provided the list of possible edges to be added is known in advance and
their simultaneous addition results in a graph in the class. All our results
also hold for relational structures and are based on the seminal result of
Nesetril and Ossona de Mendez on the existence of low tree-depth colorings
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