195 research outputs found
On 1-bend Upward Point-set Embeddings of -digraphs
We study the upward point-set embeddability of digraphs on one-sided convex
point sets with at most 1 bend per edge. We provide an algorithm to compute a
1-bend upward point-set embedding of outerplanar -digraphs on arbitrary
one-sided convex point sets. We complement this result by proving that for
every there exists a -outerplanar -digraph with
vertices and a one-sided convex point set so that does not admit a
1-bend upward point-set embedding on
Finite Embeddability of Sets and Ultrafilters
A set A of natural numbers is finitely embeddable in another such set B if
every finite subset of A has a rightward translate that is a subset of B. This
notion of finite embeddability arose in combinatorial number theory, but in
this paper we study it in its own right. We also study a related notion of
finite embeddability of ultrafilters on the natural numbers. Among other
results, we obtain connections between finite embeddability and the algebraic
and topological structure of the Stone-Cech compactification of the discrete
space of natural numbers. We also obtain connections with nonstandard models of
arithmetic.Comment: to appear in Bulletin of the Polish Academy of Sciences, Math Serie
Upward Point-Set Embeddability
We study the problem of Upward Point-Set Embeddability, that is the problem
of deciding whether a given upward planar digraph has an upward planar
embedding into a point set . We show that any switch tree admits an upward
planar straight-line embedding into any convex point set. For the class of
-switch trees, that is a generalization of switch trees (according to this
definition a switch tree is a -switch tree), we show that not every
-switch tree admits an upward planar straight-line embedding into any convex
point set, for any . Finally we show that the problem of Upward
Point-Set Embeddability is NP-complete
Embedding Four-directional Paths on Convex Point Sets
A directed path whose edges are assigned labels "up", "down", "right", or
"left" is called \emph{four-directional}, and \emph{three-directional} if at
most three out of the four labels are used. A \emph{direction-consistent
embedding} of an \mbox{-vertex} four-directional path on a set of
points in the plane is a straight-line drawing of where each vertex of
is mapped to a distinct point of and every edge points to the direction
specified by its label. We study planar direction-consistent embeddings of
three- and four-directional paths and provide a complete picture of the problem
for convex point sets.Comment: 11 pages, full conference version including all proof
Computing upward topological book embeddings of upward planar digraphs
This paper studies the problem of computing an upward topological book embedding of an upward planar digraph G, i.e. a topological book embedding of G where all edges are monotonically increasing in the upward direction. Besides having its own inherent interest in the theory of upward book embeddability, the question has applications to well studied research topics of computational geometry and of graph drawing. The main results of the paper are as follows. -Every upward planar digraph G with n vertices admits an upward topological book embedding such that every edge of G crosses the spine of the book at most once. -Every upward planar digraph G with n vertices admits a point-set embedding on any set of n distinct points in the plane such that the drawing is upward and every edge of G has at most two bends. -Every pair of upward planar digraphs sharing the same set of n vertices admits an upward simultaneous embedding with at most two bends per edge
Tight Bounds for Maximal Identifiability of Failure Nodes in Boolean Network Tomography
We study maximal identifiability, a measure recently introduced in Boolean
Network Tomography to characterize networks' capability to localize failure
nodes in end-to-end path measurements. We prove tight upper and lower bounds on
the maximal identifiability of failure nodes for specific classes of network
topologies, such as trees and -dimensional grids, in both directed and
undirected cases. We prove that directed -dimensional grids with support
have maximal identifiability using monitors; and in the
undirected case we show that monitors suffice to get identifiability of
. We then study identifiability under embeddings: we establish relations
between maximal identifiability, embeddability and graph dimension when network
topologies are model as DAGs. Our results suggest the design of networks over
nodes with maximal identifiability using
monitors and a heuristic to boost maximal identifiability on a given network by
simulating -dimensional grids. We provide positive evidence of this
heuristic through data extracted by exact computation of maximal
identifiability on examples of small real networks
Linear extensions of partial orders and Reverse Mathematics
We introduce the notion of \tau-like partial order, where \tau is one of the
linear order types \omega, \omega*, \omega+\omega*, and \zeta. For example,
being \omega-like means that every element has finitely many predecessors,
while being \zeta-like means that every interval is finite. We consider
statements of the form "any \tau-like partial order has a \tau-like linear
extension" and "any \tau-like partial order is embeddable into \tau" (when
\tau\ is \zeta\ this result appears to be new). Working in the framework of
reverse mathematics, we show that these statements are equivalent either to
B\Sigma^0_2 or to ACA_0 over the usual base system RCA_0.Comment: 8 pages, minor changes suggested by referee. To appear in MLQ -
Mathematical Logic Quarterl
The descriptive set-theoretical complexity of the embeddability relation on models of large size
We show that if \kappa\ is a weakly compact cardinal then the embeddability
relation on (generalized) trees of size \kappa\ is invariantly universal. This
means that for every analytic quasi-order R on the generalized Cantor space
2^\kappa\ there is an L_{\kappa^+ \kappa}-sentence \phi\ such that the
embeddability relation on its models of size \kappa, which are all trees, is
Borel bireducible (and, in fact, classwise Borel isomorphic) to R. In
particular, this implies that the relation of embeddability on trees of size
\kappa\ is complete for analytic quasi-orders. These facts generalize analogous
results for \kappa=\omega\ obtained in [LR05, FMR11], and it also partially
extends a result from [Bau76] concerning the structure of the embeddability
relation on linear orders of size \kappa.Comment: 41 pages, revised version, accepted for publication on the Annals of
Pure and Applied Logic. Corrected an inaccuracy in the definition of analytic
subsets of standard Borel kappa-spaces (thanks to P. Luecke and P. Schlicht
for pointing it out
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