Planare Graphen und ihre Dualgraphen auf Zylinderoberflächen

In this thesis, we investigates plane drawings of undirected and directed graphs on cylinder surfaces. In the case of undirected graphs, the vertices are positioned on a line that is parallel to the cylinder’s axis and the edge curves must not intersect this line. We show that a plane drawing is possible if and only if the graph is a double-ended queue (deque) graph, i. e., the vertices of the graph can be processed according to a linear order and the edges correspond to items in the deque inserted and removed at their end vertices. A surprising consequence resulting from these observations is that the deque characterizes planar graphs with a Hamiltonian path. This result extends the known characterization of planar graphs with a Hamiltonian cycle by two stacks. By these insights, we also obtain a new characterization of queue graphs and their duals. We also consider the complexity of deciding whether a graph is a deque graph and prove that it is NP-complete. By introducing a split operation, we obtain the splittable deque and show that it characterizes planarity. For the proof, we devise an algorithm that uses the splittable deque to test whether a rotation system is planar. In the case of directed graphs, we study upward plane drawings where the edge curves follow the direction of the cylinder’s axis (standing upward planarity; SUP) or they wind around the axis (rolling upward planarity; RUP). We characterize RUP graphs by means of their duals and show that RUP and SUP swap their roles when considering a graph and its dual. There is a physical interpretation underlying this characterization: A SUP graph is to its RUP dual graph as electric current passing through a conductor to the magnetic field surrounding the conductor. Whereas testing whether a graph is RUP is NP-hard in general [Bra14], for directed graphs without sources and sink, we develop a linear-time recognition algorithm that is based on our dual graph characterization of RUP graphs.Die Arbeit beschäftigt sich mit planaren Zeichnungen ungerichteter und gerichteter Graphen auf Zylinderoberflächen. Im ungerichteten Fall werden Zeichnungen betrachtet, bei denen die Knoten auf einer Linie parallel zur Zylinderachse positioniert werden und die Kanten diese Linie nicht schneiden dürfen. Es kann gezeigt werden, dass eine planare Zeichnung genau dann möglich ist, wenn die Kanten des Graphen in einer double-ended queue (Deque) verarbeitet werden können. Ebenso lassen sich dadurch Queue, Stack und Doppelstack charakterisieren. Eine überraschende Konsequenz aus diesen Erkenntnissen ist, dass die Deque genau die planaren Graphen mit Hamiltonpfad charakterisiert. Dies erweitert die bereits bekannte Charakterisierung planarer Graphen mit Hamiltonkreis durch den Doppelstack. Im gerichteten Fall müssen die Kantenkurven entweder in Richtung der Zylinderachse verlaufen (SUP-Graphen) oder sich um die Achse herumbewegen (RUP-Graphen). Die Arbeit charakterisiert RUP-Graphen und zeigt, dass RUP und SUP ihre Rollen tauschen, wenn man Graph und Dualgraph betrachtet. Der SUP-Graph verhält sich dabei zum RUP-Graphen wie elektrischer Strom durch einen Leiter zum induzierten Magnetfeld. Ausgehend von dieser Charakterisierung ist es möglich einen Linearzeit-Algorithmus zu entwickeln, der entscheidet ob ein gerichteter Graph ohne Quellen und Senken ein RUP-Graph ist, während der allgemeine Fall NP-hart ist [Bra14]

Operationalising assortment in the theory of marketing systems: an example from the Australian tourism marketing system for international visitors 1999-2001

Although the term “assortment” is generally known in the retail and marketing literature as the overall collection of products and services provided by a retailer, its interpretations and definitions are diverse and narrow at the same time. The aim of this thesis is to conceptualize and measure assortments in the marketing system. This particular perspective provides a generalizable context for the application of the assortment concept, and at the same time opens a door for understanding the marketing systems through the lens of assortments. The second aim of the thesis, which is instrumental in achieving the first, is to identify and measure properties of acquired assortments. Acquired assortments are the main type of assortments that represent the demand side at the interface of exchange in the marketing system. Two groups, altogether five properties of acquired assortments have been identified in the thesis. The first group involves the concept of diversity and comprises of three components: variety, balance and disparity. The second group deals with two relational properties, association and sequence. To substantiate the proposed measures, using the International Visitor Survey data, the thesis examines destination assortments acquired by short-term international visitors in the tourism marketing system of Australia in a two-year period between 1999 and 2001. Besides the theoretical implications as described by the two aims, this study also provides some useful insights for members of the Australian tourism industry in understanding how international visitors respond to the destinations offered in the system according to their own characteristics and trip characteristics together with the influence of a big event, the 2000 Sydney Olympics. Finally, nature and inter-relationships of the measures are discussed. The results show that the proposed measures reflect inter-related but different aspects of the acquired assortment. In particular, empirical inter-relationships exist between measures at the same level of categorization as well as that between different levels of aggregation. The evidence supports the idea of internal consistency in the assortment and has implications for understanding and evaluating the functioning of the marketing system