Computational Complexity of Geometric Symmetry Detection in Graphs

Constructing a visually informative drawing of an abstract graph is a problem of considerable practical importance, and has recently been the focus of much investigation. Displaying symmetry has emerged as one of the foremost criteria for achieving good drawings. Linear-time algorithms are already known for the detection and display of symmetry in trees, outerplanar graphs, and embedded planar graphs. The central results of this paper show that for general graphs, however, detecting the presence of even a single axial or rotational symmetry is NP-complete. A number of related results are also established, including the #P-completeness of counting the axial or rotational symmetries of a graph

A Better Way to Construct Tensegrities: Planar Embeddings Inform Tensegrity Assembly

Although seemingly simple, tensegrity structures are complex in nature which makes them both ideal for use in robotics and difficult to construct. We work to develop a protocol for constructing tensegrities more easily. We consider attaching a tensegrity\u27s springs to the appropriate locations on some planar arrangement of attached struts. Once all of the elements of the structure are connected, we release the struts and allow the tensegrity to find its equilibrium position. This will allow for more rapid tensegrity construction. We develop a black-box that given some tensegrity returns a flat-pack, or the information needed to perform this physical construction

Testing Planarity of Geometric Automorphisms in Linear Time

It is a well-known result that testing a graph for planarity and, in the affirmative case, computing a planar embedding can be done in linear time. In this paper, we show that the same holds if additionally we require that the produced drawing be symmetric with respect to a given automorphism of the graph. This problem arises naturally in the area of automatic graph drawing, where symmetric and planar drawings are desired whenever possible

Applications of Graph Embedding in Mesh Untangling

The subject of this thesis is mesh untangling through graph embedding, a method of laying out graphs on a planar surface, using an algorithm based on the work of Fruchterman and Reingold[1]. Meshes are a variety of graph used to represent surfaces with a wide number of applications, particularly in simulation and modelling. In the process of simulation, simulated forces can tangle the mesh through deformation and stress. The goal of this thesis was to create a tool to untangle structured meshes of complicated shapes and surfaces, including meshes with holes or concave sides. The goals of graph embedding, such as minimizing edge crossings align very well with the objectives of mesh untangling. I have designed and tested a tool which I named MUT (Mesh Untangling Tool) on meshes of various types including triangular, polygonal, and hybrid meshes. Previous methods of mesh untangling have largely been numeric or optimizationbased. Additionally, most untangling methods produce low quality graphs which must be smoothed separately to produce good meshes. Currently graph embedding techniques have only been used for smoothing of untangled meshes. I have developed a tool based on the Fruchterman-Reingold algorithm for force-directed layout[1] that effectively untangles and smooths meshes simultaneously using graph embedding techniques. It can untangle complicated meshes with irregular polygonal frames, internal holes, and other complications that previous methods struggle with. The MUT does this by using several different approaches: untangling the mesh in stages from the frame in and anchoring the mesh at corner points to stabilize the untangling

Extending constrained hierarchical layout for drawing UML activity diagrams

Ankara : The Department of Computer Engineering and Institute Engineering and Science of Bilkent University, 2002.Thesis (Master's) -- Bilkent University, 2002.Includes bibliographical references leaves 48-51.While modeling an object-oriented software, a visual language called Unified Modeling Language (UML) may be used. UML is a language and notation for specification, construction, visualization, and documentation of models of software systems. It consists of a variety of diagrams including class diagrams and activity diagrams. Graph layout has become an important area of research in Computer Science for the last couple of decades. There is a wide range of applications for graph layout including data structures, databases, software engineering, VLSI technology, electrical engineering, production planning, chemistry, and biology. Diagrams are more effective means of expressing relational information and automatic graph layout makes them to be more comprehensible. In other words, with graph layout techniques, the readability and the comprehensibility of the graphs increases and the complexity is reduced. UML diagrams are no exception. In this thesis, we present graph layout algorithms for UML activity diagrams based on constrained hierarchical layout. We use an existing implementation of constrained hierarchical layout to draw UML activity diagrams. We analyze and present the results of these new layout algorithms.Yüksel, H MehmetM.S