50 research outputs found
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
Linear rank-width of distance-hereditary graphs I. A polynomial-time algorithm
Linear rank-width is a linearized variation of rank-width, and it is deeply
related to matroid path-width. In this paper, we show that the linear
rank-width of every -vertex distance-hereditary graph, equivalently a graph
of rank-width at most , can be computed in time , and a linear layout witnessing the linear rank-width can be computed with
the same time complexity. As a corollary, we show that the path-width of every
-element matroid of branch-width at most can be computed in time
, provided that the matroid is given by an
independent set oracle.
To establish this result, we present a characterization of the linear
rank-width of distance-hereditary graphs in terms of their canonical split
decompositions. This characterization is similar to the known characterization
of the path-width of forests given by Ellis, Sudborough, and Turner [The vertex
separation and search number of a graph. Inf. Comput., 113(1):50--79, 1994].
However, different from forests, it is non-trivial to relate substructures of
the canonical split decomposition of a graph with some substructures of the
given graph. We introduce a notion of `limbs' of canonical split
decompositions, which correspond to certain vertex-minors of the original
graph, for the right characterization.Comment: 28 pages, 3 figures, 2 table. A preliminary version appeared in the
proceedings of WG'1
Tree-width for first order formulae
We introduce tree-width for first order formulae \phi, fotw(\phi). We show
that computing fotw is fixed-parameter tractable with parameter fotw. Moreover,
we show that on classes of formulae of bounded fotw, model checking is fixed
parameter tractable, with parameter the length of the formula. This is done by
translating a formula \phi\ with fotw(\phi)<k into a formula of the k-variable
fragment L^k of first order logic. For fixed k, the question whether a given
first order formula is equivalent to an L^k formula is undecidable. In
contrast, the classes of first order formulae with bounded fotw are fragments
of first order logic for which the equivalence is decidable.
Our notion of tree-width generalises tree-width of conjunctive queries to
arbitrary formulae of first order logic by taking into account the quantifier
interaction in a formula. Moreover, it is more powerful than the notion of
elimination-width of quantified constraint formulae, defined by Chen and Dalmau
(CSL 2005): for quantified constraint formulae, both bounded elimination-width
and bounded fotw allow for model checking in polynomial time. We prove that
fotw of a quantified constraint formula \phi\ is bounded by the
elimination-width of \phi, and we exhibit a class of quantified constraint
formulae with bounded fotw, that has unbounded elimination-width. A similar
comparison holds for strict tree-width of non-recursive stratified datalog as
defined by Flum, Frick, and Grohe (JACM 49, 2002).
Finally, we show that fotw has a characterization in terms of a cops and
robbers game without monotonicity cost
On Testability of First-Order Properties in Bounded-Degree Graphs and Connections to Proximity-Oblivious Testing
We study property testing of properties that are definable in first-order
logic (FO) in the bounded-degree graph and relational structure models. We show
that any FO property that is defined by a formula with quantifier prefix
is testable (i.e., testable with constant query
complexity), while there exists an FO property that is expressible by a formula
with quantifier prefix that is not testable. In the dense
graph model, a similar picture is long known (Alon, Fischer, Krivelevich,
Szegedy, Combinatorica 2000), despite the very different nature of the two
models. In particular, we obtain our lower bound by an FO formula that defines
a class of bounded-degree expanders, based on zig-zag products of graphs. We
expect this to be of independent interest.
We then use our class of FO definable bounded-degree expanders to answer a
long-standing open problem for proximity-oblivious testers (POTs). POTs are a
class of particularly simple testing algorithms, where a basic test is
performed a number of times that may depend on the proximity parameter, but the
basic test itself is independent of the proximity parameter. In their seminal
work, Goldreich and Ron [STOC 2009; SICOMP 2011] show that the graph properties
that are constant-query proximity-oblivious testable in the bounded-degree
model are precisely the properties that can be expressed as a generalised
subgraph freeness (GSF) property that satisfies the non-propagation condition.
It is left open whether the non-propagation condition is necessary. We give a
negative answer by showing that our property is a GSF property which is
propagating. Hence in particular, our property does not admit a POT. For this
result we establish a new connection between FO properties and GSF-local
properties via neighbourhood profiles.Comment: Preliminary version of this article appeared in SODA'21
(arXiv:2008.05800) and CCC'21 (arXiv:2105.08490