004 Datenverarbeitung; Informatik
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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.
The Sugiyama framework proposed in the seminal paper of 1981 is one of the most important algorithms in graph drawing and is widely used for visualizing directed graphs. In its common version, it draws graphs hierarchically and, hence, maps the topological direction to a geometric direction. However, such a hierarchical layout is not possible if the graph contains cycles, which have to be destroyed in a preceding step. In certain application and problem settings, e.g., bio sciences or periodic scheduling problems, it is important that the cyclic structure of the input graph is preserved and clearly visible in drawings. Sugiyama et al. also suggested apart from the nowadays standard horizontal algorithm a cyclic version they called recurrent hierarchies. However, this cyclic drawing style has not received much attention since. In this thesis we consider such cyclic drawings and investigate the Sugiyama framework for this new scenario. As our goal is to visualize cycles directly, the first phase of the Sugiyama framework, which is concerned with removing such cycles, can be neglected. The cyclic structure of the graph leads to new problems in the remaining phases, however, for which solutions are proposed in this thesis. The aim is a complete adaption of the Sugiyama framework for cyclic drawings. To complement our adaption of the Sugiyama framework, we also treat the problem of cyclic level planarity and present a linear time cyclic level planarity testing and embedding algorithm for strongly connected graphs.
Clustered graphs are an enhanced graph model with a recursive clustering of the vertices according to a given nesting relation. This prime technique for expressing coherence of certain parts of the graph is used in many applications, such as biochemical pathways and UML class diagrams. For directed clustered graphs usually level drawings are used, leading to clustered level graphs. In this thesis we analyze the interrelation of clusters and levels and their influence on edge crossings and cluster/edge crossings.