BAR k-VISIBILITY GRAPHS

Die selbständige und eigenhändige Anfertigung versichere ich an Eides statt. Conclusion 71 Bibliography 75 Introduction " The true mystery of the world is the visible, not the invisible. " Oscar Wilde How can geometric settings be encoded combinatorially? Which discrete structures extract the essential information from continuous arrangements? How can the combinatorial properties of visibility arrangements in the plane be captured? Questions such as these are studied in computational geometry and geometric graph theory, two fundamental areas of modern discrete mathematics. Although visibility in the plane is a very natural concept, many fundamental problems remain unsolved. Visibility graphs are a much studied approach to these problems. Generally speaking, a visibility graph consists of a set of shapes in the plane, the vertices, and a concept of visibility that defines the edges of the graph. The presented diploma thesis provides answers to the initial questions posed for the class of bar k-visibility graphs. These graphs are a structure modeling some of the essential properties of visibility configurations. We will see that they carry a lot of structure, but their combinatorics can still be quite complex. Bar visibility graphs are among the best understood classes of visibility graphs. Here the vertices correspond to horizontal line segments called bars, and visibility runs vertically along lines of sight which connect two bars while being disjoint from all others. These graphs have been completely characterized by Tamassia and Tollis [30] and independently by Wismath [33]. The concept of bar visibility graphs came up in the early 1980s when many new problems in visibility theory arised, originally inspired by applications dealing with determining visibilities between different electrical components (codeword 'VLSI-design'). Other applications arise when large graphs are to be displayed in a transparent way, and in the rapidly developing field of computer graphics. Several variations and generalizations of bar visibility graphs have been considered , using different definitions for the type of bars or the kind of visibility or often both. For example, Bose, Dean, Hutchinson and Shermer [3] introduced rectangle visibility graphs, considering rectangles with horizontal and vertical visibility. Hutchinson [21] investigates arc-and circle-visibility graphs, where the 1 2 INTRODUCTION vertices correspond to arcs of concentric circles and visibility can go through the origin. Recently, new classes of bar visibility have been introduced by restricting the vertex representations to unit bars [10] or generalizing them to sets of several bars [4]. Bar k-visibility graphs are …