Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure

Calculating the energy spectra of magnetic molecules: application of real- and spin-space symmetries

The determination of the energy spectra of small spin systems as for instance given by magnetic molecules is a demanding numerical problem. In this work we review numerical approaches to diagonalize the Heisenberg Hamiltonian that employ symmetries; in particular we focus on the spin-rotational symmetry SU(2) in combination with point-group symmetries. With these methods one is able to block-diagonalize the Hamiltonian and thus to treat spin systems of unprecedented size. In addition it provides a spectroscopic labeling by irreducible representations that is helpful when interpreting transitions induced by Electron Paramagnetic Resonance (EPR), Nuclear Magnetic Resonance (NMR) or Inelastic Neutron Scattering (INS). It is our aim to provide the reader with detailed knowledge on how to set up such a diagonalization scheme.Comment: 29 pages, many figure

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Generalized Colorings of Graphs

A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1), confirming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We define the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive different colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique

Automatic segmentation and reconstruction of traffic accident scenarios from mobile laser scanning data

Virtual reconstruction of historic sites, planning of restorations and attachments of new building parts, as well as forest inventory are few examples of fields that benefit from the application of 3D surveying data. Originally using 2D photo based documentation and manual distance measurements, the 3D information obtained from multi camera and laser scanning systems realizes a noticeable improvement regarding the surveying times and the amount of generated 3D information. The 3D data allows a detailed post processing and better visualization of all relevant spatial information. Yet, for the extraction of the required information from the raw scan data and for the generation of useable visual output, time-consuming, complex user-based data processing is still required, using the commercially available 3D software tools. In this context, the automatic object recognition from 3D point cloud and depth data has been discussed in many different works. The developed tools and methods however, usually only focus on a certain kind of object or the detection of learned invariant surface shapes. Although the resulting methods are applicable for certain practices of data segmentation, they are not necessarily suitable for arbitrary tasks due to the varying requirements of the different fields of research. This thesis presents a more widespread solution for automatic scene reconstruction from 3D point clouds, targeting street scenarios, specifically for the task of traffic accident scene analysis and documentation. The data, obtained by sampling the scene using a mobile scanning system is evaluated, segmented, and finally used to generate detailed 3D information of the scanned environment. To realize this aim, this work adapts and validates various existing approaches on laser scan segmentation regarding the application on accident relevant scene information, including road surfaces and markings, vehicles, walls, trees and other salient objects. The approaches are therefore evaluated regarding their suitability and limitations for the given tasks, as well as for possibilities concerning the combined application together with other procedures. The obtained knowledge is used for the development of new algorithms and procedures to allow a satisfying segmentation and reconstruction of the scene, corresponding to the available sampling densities and precisions. Besides the segmentation of the point cloud data, this thesis presents different visualization and reconstruction methods to achieve a wider range of possible applications of the developed system for data export and utilization in different third party software tools

Was The Buddha a Reductionist About The Self?

This paper examines whether a reductionist view of the self can be found in the Suttas of early Buddhism. I will argue that the views of the self exemplified in the Path of Purification (Visuddhimagga) and the Abhidhamma Commentaries are similar to reductionist views of the self put forward by Western philosophers such as David Hume and Derek Parfit. I shall argue that the views of the Visuddhimagga can be seen as a legitimate development of the ideas of the Suttas. I will also argue however that an opposing view, namely the rejection of a 'realist semantics', can be found in nascent form in the Suttas. I will demonstrate that, when legitimately developed, this view can be seen to contradict the reductionist view of the self