289 research outputs found

    Planar Open Rectangle-of-Influence Drawings

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    A straight line drawing of a graph is an open weak rectangle-of-influence (RI) drawing, if there is no vertex in the relative interior of the axis parallel rectangle induced by the end points of each edge. Despite recent interest of the graph drawing community in rectangle-of-influence drawings, no algorithm is known to test whether a graph has a planar open weak RI-drawing, not even for inner triangulated graphs. In this thesis, we have two major contributions. First we study open weak RI-drawings of plane graphs that must have a non-aligned frame, i.e., the graph obtained from removing the interior of every filled triangle is drawn such that no two vertices have the same coordinate. We introduce a new way to assign labels to angles, i.e., instances of vertices on faces. Using this labeling, we provide necessary and sufficient conditions characterizing those plane graphs that have open weak RI-drawings with non-aligned frame. We also give a polynomial algorithm to construct such a drawing if one exists. Our second major result is a negative result: deciding if a planar graph (i.e., one where we can choose the planar embedding) has an open weak RI-drawing is NP-complete. NP-completeness holds even for open weak RI-drawings with non-aligned frames

    Drawing graphs for cartographic applications

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    Graph Drawing is a relatively young area that combines elements of graph theory, algorithms, (computational) geometry and (computational) topology. Research in this field concentrates on developing algorithms for drawing graphs while satisfying certain aesthetic criteria. These criteria are often expressed in properties like edge complexity, number of edge crossings, angular resolutions, shapes of faces or graph symmetries and in general aim at creating a drawing of a graph that conveys the information to the reader in the best possible way. Graph drawing has applications in a wide variety of areas which include cartography, VLSI design and information visualization. In this thesis we consider several graph drawing problems. The first problem we address is rectilinear cartogram construction. A cartogram, also known as value-by-area map, is a technique used by cartographers to visualize statistical data over a set of geographical regions like countries, states or counties. The regions of a cartogram are deformed such that the area of a region corresponds to a particular geographic variable. The shapes of the regions depend on the type of cartogram. We consider rectilinear cartograms of constant complexity, that is cartograms where each region is a rectilinear polygon with a constant number of vertices. Whether a cartogram is good is determined by how closely the cartogram resembles the original map and how precisely the area of its regions describe the associated values. The cartographic error is defined for each region as jAc¡Asj=As, where Ac is the area of the region in the cartogram and As is the specified area of that region, given by the geographic variable to be shown. In this thesis we consider the construction of rectilinear cartograms that have correct adjacencies of the regions and zero cartographic error. We show that any plane triangulated graph admits a rectilinear cartogram where every region has at most 40 vertices which can be constructed in O(nlogn) time. We also present experimental results that show that in practice the algorithm works significantly better than suggested by the complexity bounds. In our experiments on real-world data we were always able to construct a cartogram where the average number of vertices per region does not exceed five. Since a rectangle has four vertices, this means that most of the regions of our rectilinear car tograms are in fact rectangles. Moreover, the maximum number vertices of each region in these cartograms never exceeded ten. The second problem we address in this thesis concerns cased drawings of graphs. The vertices of a drawing are commonly marked with a disk, but differentiating between vertices and edge crossings in a dense graph can still be difficult. Edge casing is a wellknown method—used, for example, in electrical drawings, when depicting knots, and, more generally, in information visualization—to alleviate this problem and to improve the readability of a drawing. A cased drawing orders the edges of each crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. One can also envision that every edge is encased in a strip of the background color and that the casing of the upper edge covers the lower edge at the crossing. If there are no application-specific restrictions that dictate the order of the edges at each crossing, then we can in principle choose freely how to arrange them. However, certain orders will lead to a more readable drawing than others. In this thesis we formulate aesthetic criteria for a cased drawing as optimization problems and solve these problems. For most of the problems we present either a polynomial time algorithm or demonstrate that the problem is NP-hard. Finally we consider a combinatorial question in computational topology concerning three types of objects: closed curves in the plane, surfaces immersed in the plane, and surfaces embedded in space. In particular, we study casings of closed curves in the plane to decide whether these curves can be embedded as the boundaries of certain special surfaces. We show that it is NP-complete to determine whether an immersed disk is the projection of a surface embedded in space, or whether a curve is the boundary of an immersed surface in the plane that is not constrained to be a disk. However, when a casing is supplied with a self-intersecting curve, describing which component of the curve lies above and which below at each crossing, we can determine in time linear in the number of crossings whether the cased curve forms the projected boundary of a surface in space. As a related result, we show that an immersed surface with a single boundary curve that crosses itself n times has at most 2n=2 combinatorially distinct spatial embeddings and we discuss the existence of fixed-parameter tractable algorithms for related problems

    Drawing Planar Graphs with Few Geometric Primitives

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    We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let nn denote the number of vertices of a graph. We show that trees can be drawn with 3n/43n/4 straight-line segments on a polynomial grid, and with n/2n/2 straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with (8n−17)/3(8n-17)/3 segments on an O(n)×O(n2)O(n)\times O(n^2) grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with 3n/23n/2 edges on an O(n)×O(n2)O(n)\times O(n^2) grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only (5n−11)/3(5n - 11)/3 arcs. This is significantly smaller than the lower bound of 2n2n for line segments for a nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2017

    4-labelings and grid embeddings of plane quadrangulations

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    AbstractA straight-line drawing of a planar graph G is a closed rectangle-of-influence drawing if for each edge uv, the closed axis-parallel rectangle with opposite corners u and v contains no other vertices. We show that each quadrangulation on n vertices has a closed rectangle-of-influence drawing on the (n−3)×(n−3) grid.The algorithm is based on angle labeling and simple face counting in regions. This answers the question of what would be a grid embedding of quadrangulations analogous to Schnyder’s classical algorithm for embedding triangulations and extends previous results on book embeddings for quadrangulations from Felsner, Huemer, Kappes, and Orden.A further compaction step yields a straight-line drawing of a quadrangulation on the (⌈n2⌉−1)×(⌈3n4⌉−1) grid. The advantage over other existing algorithms is that it is not necessary to add edges to the quadrangulation to make it 4-connected

    Schematics of Graphs and Hypergraphs

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    Graphenzeichnen als ein Teilgebiet der Informatik befasst sich mit dem Ziel Graphen oder deren Verallgemeinerung Hypergraphen geometrisch zu realisieren. BeschrĂ€nkt man sich dabei auf visuelles Hervorheben von wesentlichen Informationen in Zeichenmodellen, spricht man von Schemata. Hauptinstrumente sind Konstruktionsalgorithmen und Charakterisierungen von Graphenklassen, die fĂŒr die Konstruktion geeignet sind. In dieser Arbeit werden Schemata fĂŒr Graphen und Hypergraphen formalisiert und mit den genannten Instrumenten untersucht. In der Dissertation wird zunĂ€chst das „partial edge drawing“ (kurz: PED) Modell fĂŒr Graphen (bezĂŒglich gradliniger Zeichnung) untersucht. Dabei wird um Kreuzungen im Zentrum der Kante visuell zu eliminieren jede Kante durch ein kreuzungsfreies TeilstĂŒck (= Stummel) am Start- und am Zielknoten ersetzt. Als Standard hat sich eine PED-Variante etabliert, in der das LĂ€ngenverhĂ€ltnis zwischen Stummel und Kante genau 1⁄4 ist (kurz: 1⁄4-SHPED). FĂŒr 1⁄4-SHPEDs werden Konstruktionsalgorithmen, Klassifizierung, Implementierung und Evaluation prĂ€sentiert. Außerdem werden PED-Varianten mit festen Knotenpositionen und auf Basis orthogonaler Zeichnungen erforscht. Danach wird das BUS Modell fĂŒr Hypergraphen untersucht, in welchem Hyperkanten durch fette horizontale oder vertikale – als BUS bezeichnete – Segmente reprĂ€sentiert werden. Dazu wird eine vollstĂ€ndige Charakterisierung von planaren Inzidenzgraphen von Hypergraphen angegeben, die eine planare Zeichnung im BUS Modell besitzen, und diverse planare BUS-Varianten mit festen Knotenpositionen werden diskutiert. Zum Schluss wird erstmals eine Punktmenge von subquadratischer GrĂ¶ĂŸe angegeben, die eine planare Einbettung (Knoten werden auf Punkte abgebildet) von 2-außenplanaren Graphen ermöglicht

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Restricted String Representations

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    A string representation of a graph assigns to every vertex a curve in the plane so that two curves intersect if and only if the represented vertices are adjacent. This work investigates string representations of graphs with an emphasis on the shapes of curves and the way they intersect. We strengthen some previously known results and show that every planar graph has string representations where every curve consists of axis-parallel line segments with at most two bends (those are the so-called B2B_2-VPG representations) and simultaneously two curves intersect each other at most once (those are the so-called 1-string representations). Thus, planar graphs are B2B_2-VPG 11-string graphs. We further show that with some restrictions on the shapes of the curves, string representations can be used to produce approximation algorithms for several hard problems. The B2B_2-VPG representations of planar graphs satisfy these restrictions. We attempt to further restrict the number of bends in VPG representations for subclasses of planar graphs, and investigate B1B_1-VPG representations. We propose new classes of string representations for planar graphs that we call ``order-preserving.'' Order-preservation is an interesting property which relates the string representation to the planar embedding of the graph, and we believe that it might prove useful when constructing string representations. Finally, we extend our investigation of string representations to string representations that require some curves to intersect multiple times. We show that there are outer-string graphs that require an exponential number of crossings in their outer-string representations. Our construction also proves that 1-planar graphs, i.e., graphs that are no longer planar, yet fairly close to planar graphs, may have string representations, but they are not always 1-string

    Modified Bistable Modules for Bias Deployable Structures

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    Financiado para publicación en acceso aberto: Universidade da Coruña/CISUG[Abstract] Bias deployable grids are meshes with two directions of rotation on the ground plan with respect to the edges. They offer benefits such as three-dimensional resistance with supports around the entire perimeter of a rectangular layout, and consist exclusively of load-bearing scissors as opposed to the usual combinations of load-bearing scissors and bracing scissors. However, their resistance to angular distortion is limited, and they require auxiliary elements to maintain the fully deployed position. Nevertheless, they are very promising solutions for medium-span emergency buildings. This paper proposes a bistable module adapted to bias deployable structures. The geometrical incompatibilities of several modules are analysed together with their behaviour based on the kinematic models that were built, which alternate different types of nodes and different geometries of the perimeter scissors, making it possible to calibrate the level of incompatibility introduced. The dimensions of the nodes are also taken into account. The tests are checked against the results of several series of dynamic calculations.This research was carried out as a part of the Spanish Research Project on Deployable and Modular Constructions for Situations of Humanitarian Catastrophe, CODEMOSCH (Reference BIA2016-79459-R), funded by the Spanish Ministry of Industry, Energy, and Competitiveness (MINECO). Financing of the open access fee: Universidade da Coruña / CISU
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