127 research outputs found
Exploiting Air-Pressure to Map Floorplans on Point Sets
We prove a conjecture of Ackerman, Barequet and Pinter. Every floorplan with n segments can be embedded on every set of n points in generic position. The construction makes use of area universal floorplans also known as area universal rectangular layouts.
The notion of area used in our context depends on a nonuniform density function. We, therefore, have to generalize the theory of area universal floorplans to this situation. The method is then used to prove a result about accommodating points in floorplans that is slightly more general than the conjecture of Ackerman et al
Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem
We consider arrangements of axis-aligned rectangles in the plane. A geometric
arrangement specifies the coordinates of all rectangles, while a combinatorial
arrangement specifies only the respective intersection type in which each pair
of rectangles intersects. First, we investigate combinatorial contact
arrangements, i.e., arrangements of interior-disjoint rectangles, with a
triangle-free intersection graph. We show that such rectangle arrangements are
in bijection with the 4-orientations of an underlying planar multigraph and
prove that there is a corresponding geometric rectangle contact arrangement.
Moreover, we prove that every triangle-free planar graph is the contact graph
of such an arrangement. Secondly, we introduce the question whether a given
rectangle arrangement has a combinatorially equivalent square arrangement. In
addition to some necessary conditions and counterexamples, we show that
rectangle arrangements pierced by a horizontal line are squarable under certain
sufficient conditions.Comment: 15 pages, 13 figures, extended version of a paper to appear at the
International Symposium on Graph Drawing and Network Visualization (GD) 201
Contact Representations of Graphs in 3D
We study contact representations of graphs in which vertices are represented
by axis-aligned polyhedra in 3D and edges are realized by non-zero area common
boundaries between corresponding polyhedra. We show that for every 3-connected
planar graph, there exists a simultaneous representation of the graph and its
dual with 3D boxes. We give a linear-time algorithm for constructing such a
representation. This result extends the existing primal-dual contact
representations of planar graphs in 2D using circles and triangles. While
contact graphs in 2D directly correspond to planar graphs, we next study
representations of non-planar graphs in 3D. In particular we consider
representations of optimal 1-planar graphs. A graph is 1-planar if there exists
a drawing in the plane where each edge is crossed at most once, and an optimal
n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a
linear-time algorithm for representing optimal 1-planar graphs without
separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph
admits a representation with boxes. Hence, we consider contact representations
with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a
quadratic-time algorithm for representing optimal 1-planar graph with L-shaped
polyhedra
ERAD Components in Organisms with Complex Red Plastids Suggest Recruitment of a Preexisting Protein Transport Pathway for the Periplastid Membrane
The plastids of cryptophytes, haptophytes, and heterokontophytes (stramenopiles) (together once known as chromists) are surrounded by four membranes, reflecting the origin of these plastids through secondary endosymbiosis. They share this trait with apicomplexans, which are alveolates, the plastids of which have been suggested to stem from the same secondary symbiotic event and therefore form a phylogenetic clade, the chromalveolates. The chromists are quantitatively the most important eukaryotic contributors to primary production in marine ecosystems. The mechanisms of protein import across their four plastid membranes are still poorly understood. Components of an endoplasmic reticulum-associated degradation (ERAD) machinery in cryptophytes, partially encoded by the reduced genome of the secondary symbiont (the nucleomorph), are implicated in protein transport across the second outermost plastid membrane. Here, we show that the haptophyte Emiliania huxleyi, like cryptophytes, stramenopiles, and apicomplexans, possesses a nuclear-encoded symbiont-specific ERAD machinery (SELMA, symbiont-specific ERAD-like machinery) in addition to the host ERAD system, with targeting signals that are able to direct green fluorescent protein or yellow fluorescent protein to the predicted cellular localization in transformed cells of the stramenopile Phaeodactylum tricornutum. Phylogenies of the duplicated ERAD factors reveal that all SELMA components trace back to a red algal origin. In contrast, the host copies of cryptophytes and haptophytes associate with the green lineage to the exclusion of stramenopiles and alveolates. Although all chromalveolates with four membrane-bound plastids possess the SELMA system, this has apparently not arisen in a single endosymbiotic event. Thus, our data do not support the chromalveolate hypothesis
On Arrangements of Orthogonal Circles
In this paper, we study arrangements of orthogonal circles, that is,
arrangements of circles where every pair of circles must either be disjoint or
intersect at a right angle. Using geometric arguments, we show that such
arrangements have only a linear number of faces. This implies that orthogonal
circle intersection graphs have only a linear number of edges. When we restrict
ourselves to orthogonal unit circles, the resulting class of intersection
graphs is a subclass of penny graphs (that is, contact graphs of unit circles).
We show that, similarly to penny graphs, it is NP-hard to recognize orthogonal
unit circle intersection graphs.Comment: Appears in the Proceedings of the 27th International Symposium on
Graph Drawing and Network Visualization (GD 2019
On the Order Dimension of Convex Geometries
We study the order dimension of the lattice of closed sets for a convex
geometry. Further, we prove the existence of large convex geometries realized
by planar point sets that have very low order dimension. We show that the
planar point set of Erdos and Szekeres from 1961 which is a set of 2^(n-2)
points and contains no convex n-gon has order dimension n - 1 and any larger
set of points has order dimension strictly larger than n - 1.Comment: 12 pages, 2 figure
On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings
We study two variants of the well-known orthogonal drawing model: (i) the
smooth orthogonal, and (ii) the octilinear. Both models form an extension of
the orthogonal, by supporting one additional type of edge segments (circular
arcs and diagonal segments, respectively).
For planar graphs of max-degree 4, we analyze relationships between the graph
classes that can be drawn bendless in the two models and we also prove
NP-hardness for a restricted version of the bendless drawing problem for both
models. For planar graphs of higher degree, we present an algorithm that
produces bi-monotone smooth orthogonal drawings with at most two segments per
edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Upward Three-Dimensional Grid Drawings of Graphs
A \emph{three-dimensional grid drawing} of a graph is a placement of the
vertices at distinct points with integer coordinates, such that the straight
line segments representing the edges do not cross. Our aim is to produce
three-dimensional grid drawings with small bounding box volume. We prove that
every -vertex graph with bounded degeneracy has a three-dimensional grid
drawing with volume. This is the broadest class of graphs admiting
such drawings. A three-dimensional grid drawing of a directed graph is
\emph{upward} if every arc points up in the z-direction. We prove that every
directed acyclic graph has an upward three-dimensional grid drawing with
volume, which is tight for the complete dag. The previous best upper
bound was . Our main result is that every -colourable directed
acyclic graph ( constant) has an upward three-dimensional grid drawing with
volume. This result matches the bound in the undirected case, and
improves the best known bound from for many classes of directed
acyclic graphs, including planar, series parallel, and outerplanar
Visibility Representations of Boxes in 2.5 Dimensions
We initiate the study of 2.5D box visibility representations (2.5D-BR) where
vertices are mapped to 3D boxes having the bottom face in the plane and
edges are unobstructed lines of sight parallel to the - or -axis. We
prove that: Every complete bipartite graph admits a 2.5D-BR; The
complete graph admits a 2.5D-BR if and only if ; Every
graph with pathwidth at most admits a 2.5D-BR, which can be computed in
linear time. We then turn our attention to 2.5D grid box representations
(2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit
square at integer coordinates. We show that an -vertex graph that admits a
2.5D-GBR has at most edges and this bound is tight. Finally,
we prove that deciding whether a given graph admits a 2.5D-GBR with a given
footprint is NP-complete. The footprint of a 2.5D-BR is the set of
bottom faces of the boxes in .Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Convexity in partial cubes: the hull number
We prove that the combinatorial optimization problem of determining the hull
number of a partial cube is NP-complete. This makes partial cubes the minimal
graph class for which NP-completeness of this problem is known and improves
some earlier results in the literature.
On the other hand we provide a polynomial-time algorithm to determine the
hull number of planar partial cube quadrangulations.
Instances of the hull number problem for partial cubes described include
poset dimension and hitting sets for interiors of curves in the plane.
To obtain the above results, we investigate convexity in partial cubes and
characterize these graphs in terms of their lattice of convex subgraphs,
improving a theorem of Handa. Furthermore we provide a topological
representation theorem for planar partial cubes, generalizing a result of
Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure
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