1,360 research outputs found
Characterisations and Examples of Graph Classes with Bounded Expansion
Classes with bounded expansion, which generalise classes that exclude a
topological minor, have recently been introduced by Ne\v{s}et\v{r}il and Ossona
de Mendez. These classes are defined by the fact that the maximum average
degree of a shallow minor of a graph in the class is bounded by a function of
the depth of the shallow minor. Several linear-time algorithms are known for
bounded expansion classes (such as subgraph isomorphism testing), and they
allow restricted homomorphism dualities, amongst other desirable properties. In
this paper we establish two new characterisations of bounded expansion classes,
one in terms of so-called topological parameters, the other in terms of
controlling dense parts. The latter characterisation is then used to show that
the notion of bounded expansion is compatible with Erd\"os-R\'enyi model of
random graphs with constant average degree. In particular, we prove that for
every fixed , there exists a class with bounded expansion, such that a
random graph of order and edge probability asymptotically almost
surely belongs to the class. We then present several new examples of classes
with bounded expansion that do not exclude some topological minor, and appear
naturally in the context of graph drawing or graph colouring. In particular, we
prove that the following classes have bounded expansion: graphs that can be
drawn in the plane with a bounded number of crossings per edge, graphs with
bounded stack number, graphs with bounded queue number, and graphs with bounded
non-repetitive chromatic number. We also prove that graphs with `linear'
crossing number are contained in a topologically-closed class, while graphs
with bounded crossing number are contained in a minor-closed class
Monadic second-order definable graph orderings
We study the question of whether, for a given class of finite graphs, one can
define, for each graph of the class, a linear ordering in monadic second-order
logic, possibly with the help of monadic parameters. We consider two variants
of monadic second-order logic: one where we can only quantify over sets of
vertices and one where we can also quantify over sets of edges. For several
special cases, we present combinatorial characterisations of when such a linear
ordering is definable. In some cases, for instance for graph classes that omit
a fixed graph as a minor, the presented conditions are necessary and
sufficient; in other cases, they are only necessary. Other graph classes we
consider include complete bipartite graphs, split graphs, chordal graphs, and
cographs. We prove that orderability is decidable for the so called
HR-equational classes of graphs, which are described by equation systems and
generalize the context-free languages
Nowhere Dense Graph Classes and Dimension
Nowhere dense graph classes provide one of the least restrictive notions of
sparsity for graphs. Several equivalent characterizations of nowhere dense
classes have been obtained over the years, using a wide range of combinatorial
objects. In this paper we establish a new characterization of nowhere dense
classes, in terms of poset dimension: A monotone graph class is nowhere dense
if and only if for every and every , posets of height
at most with elements and whose cover graphs are in the class have
dimension .Comment: v4: Minor changes suggested by a refere
A characterisation of generically rigid frameworks on surfaces of revolution
A foundational theorem of Laman provides a counting characterisation of the
finite simple graphs whose generic bar-joint frameworks in two dimensions are
infinitesimally rigid. Recently a Laman-type characterisation was obtained for
frameworks in three dimensions whose vertices are constrained to concentric
spheres or to concentric cylinders. Noting that the plane and the sphere have 3
independent locally tangential infinitesimal motions while the cylinder has 2,
we obtain here a Laman-Henneberg theorem for frameworks on algebraic surfaces
with a 1-dimensional space of tangential motions. Such surfaces include the
torus, helicoids and surfaces of revolution. The relevant class of graphs are
the (2,1)-tight graphs, in contrast to (2,3)-tightness for the plane/sphere and
(2,2)-tightness for the cylinder. The proof uses a new characterisation of
simple (2,1)-tight graphs and an inductive construction requiring generic
rigidity preservation for 5 graph moves, including the two Henneberg moves, an
edge joining move and various vertex surgery moves.Comment: 23 pages, 5 figures. Minor revisions - most importantly, the new
version has a different titl
Reconfiguration on sparse graphs
A vertex-subset graph problem Q defines which subsets of the vertices of an
input graph are feasible solutions. A reconfiguration variant of a
vertex-subset problem asks, given two feasible solutions S and T of size k,
whether it is possible to transform S into T by a sequence of vertex additions
and deletions such that each intermediate set is also a feasible solution of
size bounded by k. We study reconfiguration variants of two classical
vertex-subset problems, namely Independent Set and Dominating Set. We denote
the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete
on graphs of bounded bandwidth and W[1]-hard parameterized by k on general
graphs. We show that ISR is fixed-parameter tractable parameterized by k when
the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we
answer positively an open question concerning the parameterized complexity of
the problem on graphs of bounded treewidth. Moreover, our techniques generalize
recent results showing that ISR is fixed-parameter tractable on planar graphs
and graphs of bounded degree. For DSR, we show the problem fixed-parameter
tractable parameterized by k when the input graph does not contain large
bicliques, a class of graphs which includes graphs of bounded degeneracy and
nowhere-dense graphs
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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