746 research outputs found
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
Preservation and decomposition theorems for bounded degree structures
We provide elementary algorithms for two preservation theorems for
first-order sentences (FO) on the class \^ad of all finite structures of degree
at most d: For each FO-sentence that is preserved under extensions
(homomorphisms) on \^ad, a \^ad-equivalent existential (existential-positive)
FO-sentence can be constructed in 5-fold (4-fold) exponential time. This is
complemented by lower bounds showing that a 3-fold exponential blow-up of the
computed existential (existential-positive) sentence is unavoidable. Both
algorithms can be extended (while maintaining the upper and lower bounds on
their time complexity) to input first-order sentences with modulo m counting
quantifiers (FO+MODm). Furthermore, we show that for an input FO-formula, a
\^ad-equivalent Feferman-Vaught decomposition can be computed in 3-fold
exponential time. We also provide a matching lower bound.Comment: 42 pages and 3 figures. This is the full version of: Frederik
Harwath, Lucas Heimberg, and Nicole Schweikardt. Preservation and
decomposition theorems for bounded degree structures. In Joint Meeting of the
23rd EACSL Annual Conference on Computer Science Logic (CSL) and the 29th
Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS'14,
pages 49:1-49:10. ACM, 201
The Generalised Colouring Numbers on Classes of Bounded Expansion
The generalised colouring numbers , ,
and were introduced by Kierstead and Yang as
generalisations of the usual colouring number, also known as the degeneracy of
a graph, and have since then found important applications in the theory of
bounded expansion and nowhere dense classes of graphs, introduced by
Ne\v{s}et\v{r}il and Ossona de Mendez. In this paper, we study the relation of
the colouring numbers with two other measures that characterise nowhere dense
classes of graphs, namely with uniform quasi-wideness, studied first by Dawar
et al. in the context of preservation theorems for first-order logic, and with
the splitter game, introduced by Grohe et al. We show that every graph
excluding a fixed topological minor admits a universal order, that is, one
order witnessing that the colouring numbers are small for every value of .
Finally, we use our construction of such orders to give a new proof of a result
of Eickmeyer and Kawarabayashi, showing that the model-checking problem for
successor-invariant first-order formulas is fixed-parameter tractable on
classes of graphs with excluded topological minors
Preservation Theorems Through the Lens of Topology
In this paper, we introduce a family of topological spaces that captures the existence of preservation theorems. The structure of those spaces allows us to study the relativisation of preservation theorems under suitable definitions of surjective morphisms, subclasses, sums, products, topological closures, and projective limits. Throughout the paper, we also integrate already known results into this new framework and show how it captures the essence of their proofs
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
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)
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