584 research outputs found
Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles
We consider a coloring problem on dynamic, one-dimensional point sets: points
appearing and disappearing on a line at given times. We wish to color them with
k colors so that at any time, any sequence of p(k) consecutive points, for some
function p, contains at least one point of each color.
We prove that no such function p(k) exists in general. However, in the
restricted case in which points appear gradually, but never disappear, we give
a coloring algorithm guaranteeing the property at any time with p(k)=3k-2. This
can be interpreted as coloring point sets in R^2 with k colors such that any
bottomless rectangle containing at least 3k-2 points contains at least one
point of each color. Here a bottomless rectangle is an axis-aligned rectangle
whose bottom edge is below the lowest point of the set. For this problem, we
also prove a lower bound p(k)>ck, where c>1.67. Hence for every k there exists
a point set, every k-coloring of which is such that there exists a bottomless
rectangle containing ck points and missing at least one of the k colors.
Chen et al. (2009) proved that no such function exists in the case of
general axis-aligned rectangles. Our result also complements recent results
from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).Comment: A preliminary version was presented by a subset of the authors to the
European Workshop on Computational Geometry, held in Assisi (Italy) on March
19-21, 201
Unsplittable coverings in the plane
A system of sets forms an {\em -fold covering} of a set if every point
of belongs to at least of its members. A -fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body , there exists a constant such that every
-fold covering of the plane with translates of splits into
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every , an
unsplittable -fold covering of the plane with translates of any open convex
body which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets do not misbehave,
they satisfy the conjecture: every -fold covering of any region of the plane
by translates of such a set splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant such that, for
any positive integer , every -fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} sets
Drawing Planar Graphs with a Prescribed Inner Face
Given a plane graph (i.e., a planar graph with a fixed planar embedding)
and a simple cycle in whose vertices are mapped to a convex polygon, we
consider the question whether this drawing can be extended to a planar
straight-line drawing of . We characterize when this is possible in terms of
simple necessary conditions, which we prove to be sufficient. This also leads
to a linear-time testing algorithm. If a drawing extension exists, it can be
computed in the same running time
study protocol for two randomized pragmatic trials
Background Chronic low back pain (LBP) and neck pain (NP) are highly prevalent
conditions resulting in high economic costs. Treatment guidelines recommend
relaxation techniques, such as progressive muscle relaxation, as adjuvant
therapies. Self-care interventions could have the potential to reduce costs in
the health care system, but their effectiveness, especially in a usual care
setting, is unclear. The aim of these two pragmatic randomized studies is to
evaluate whether an additional app-delivered relaxation is more effective in
the reduction of chronic LBP or NP than usual care alone. Methods/design Each
pragmatic randomized two-armed study aims to include a total of 220 patients
aged 18 to 65 years with chronic (>12 weeks) LBP or NP and an average pain
intensity ofââ„â4 on a numeric rating scale (NRS) in the 7 days before
recruitment. The participants will be randomized into an intervention and a
usual care group. The intervention group will be instructed to practice one of
these 3 relaxation techniques on at least 5 days/week for 15 minutes/day over
a period of 6 months starting on the day of randomization: autogenic training,
mindfulness meditation, or guided imagery. Instructions and exercises will be
provided using a smartphone app, baseline information will be collected using
paper and pencil. Follow-up information (daily, weekly, and after 3 and 6
months) will be collected using electronic diaries and questionnaires included
in the app. The primary outcome measure will be the mean LBP or NP intensity
during the first 3 months of intervention based on daily pain intensity
measurements on a NRS (0â=âno pain, 10â=âworst possible pain). The secondary
outcome parameters will include the mean pain intensity during the first 6
months after randomization based on daily measurements, the mean pain
intensity measured weekly as the average pain intensity of the previous 7 days
over 3 and 6 months, pain acceptance, âLBP- and NP-relatedâ stress, sick leave
days, pain medication intake, adherence, suspected adverse reaction, and
serious adverse events. Discussion The designed studies reflect a usual self-
care setting and will provide evidence on a pragmatic self-care intervention
that is easy to combine with care provided by medical professionals
Obstacle Numbers of Planar Graphs
Given finitely many connected polygonal obstacles in the
plane and a set of points in general position and not in any obstacle, the
{\em visibility graph} of with obstacles is the (geometric)
graph with vertex set , where two vertices are adjacent if the straight line
segment joining them intersects no obstacle. The obstacle number of a graph
is the smallest integer such that is the visibility graph of a set of
points with obstacles. If is planar, we define the planar obstacle
number of by further requiring that the visibility graph has no crossing
edges (hence that it is a planar geometric drawing of ). In this paper, we
prove that the maximum planar obstacle number of a planar graph of order is
, the maximum being attained (in particular) by maximal bipartite planar
graphs. This displays a significant difference with the standard obstacle
number, as we prove that the obstacle number of every bipartite planar graph
(and more generally in the class PURE-2-DIR of intersection graphs of straight
line segments in two directions) of order at least is .Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Five Lessons Learned From Randomized Controlled Trials on Mobile Health Interventions: Consensus Procedure on Practical Recommendations for Sustainable Research
Background: Clinical research on mobile health (mHealth) interventions is too slow in comparison to the rapid speed of technological advances, thereby impeding sustainable research and evidence-based implementation of mHealth interventions.
Objective: We aimed to establish practical lessons from the experience of our working group, which might accelerate the development of future mHealth interventions and their evaluation by randomized controlled trials (RCTs).
Methods: This paper is based on group and expert discussions, and focuses on the researchersâ perspectives after four RCTson mHealth interventions for chronic pain.
Results: The following five lessons are presented, which are based on practical application, increase of speed, and sustainability:
(1) explore stakeholder opinions, (2) develop the mHealth app and trial simultaneously, (3) minimize complexity, (4) manage necessary resources, and (5) apply behavior change techniques.
Conclusions: The five lessons developed may lead toward an agile research environment. Agility might be the key factor in
the development and research process of a potentially sustainable and evidence-based mHealth intervention
Beyond Outerplanarity
We study straight-line drawings of graphs where the vertices are placed in
convex position in the plane, i.e., convex drawings. We consider two families
of graph classes with nice convex drawings: outer -planar graphs, where each
edge is crossed by at most other edges; and, outer -quasi-planar graphs
where no edges can mutually cross. We show that the outer -planar graphs
are -degenerate, and consequently that every
outer -planar graph can be -colored, and this
bound is tight. We further show that every outer -planar graph has a
balanced separator of size . This implies that every outer -planar
graph has treewidth . For fixed , these small balanced separators
allow us to obtain a simple quasi-polynomial time algorithm to test whether a
given graph is outer -planar, i.e., none of these recognition problems are
NP-complete unless ETH fails. For the outer -quasi-planar graphs we prove
that, unlike other beyond-planar graph classes, every edge-maximal -vertex
outer -quasi planar graph has the same number of edges, namely . We also construct planar 3-trees that are not outer
-quasi-planar. Finally, we restrict outer -planar and outer
-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence
on the boundary is a cycle in the graph. For each , we express closed outer
-planarity and \emph{closed outer -quasi-planarity} in extended monadic
second-order logic. Thus, closed outer -planarity is linear-time testable by
Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
-Stars or On Extending a Drawing of a Connected Subgraph
We consider the problem of extending the drawing of a subgraph of a given
plane graph to a drawing of the entire graph using straight-line and polyline
edges. We define the notion of star complexity of a polygon and show that a
drawing of an induced connected subgraph can be extended with at
most bends per edge, where is the
largest star complexity of a face of and is the size of the
largest face of . This result significantly improves the previously known
upper bound of [5] for the case where is connected. We also show
that our bound is worst case optimal up to a small additive constant.
Additionally, we provide an indication of complexity of the problem of testing
whether a star-shaped inner face can be extended to a straight-line drawing of
the graph; this is in contrast to the fact that the same problem is solvable in
linear time for the case of star-shaped outer face [9] and convex inner face
[13].Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
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