Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

Simple Wriggling is Hard unless You Are a Fat Hippo

We prove that it is NP-hard to decide whether two points in a polygonal domain with holes can be connected by a wire. This implies that finding any approximation to the shortest path for a long snake amidst polygonal obstacles is NP-hard. On the positive side, we show that snake's problem is "length-tractable": if the snake is "fat", i.e., its length/width ratio is small, the shortest path can be computed in polynomial time.Comment: A shorter version is to be presented at FUN 201

Minimum-weight triangulation is NP-hard

A triangulation of a planar point set S is a maximal plane straight-line graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths. We prove that the decision version of this problem is NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the gadgets is established with computer assistance, using dynamic programming on polygonal faces, as well as the beta-skeleton heuristic to certify that certain edges belong to the minimum-weight triangulation.Comment: 45 pages (including a technical appendix of 13 pages), 28 figures. This revision contains a few improvements in the expositio

05191 Abstracts Collection -- Graph Drawing

From 08.05.05 to 13.05.05, the Dagstuhl Seminar 05191 ``Graph Drawing\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Utilization of Decomposition Techniques for Analyzing and Characterizing Flows

This thesis presents the utilization of two different decomposition techniques, proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD), for enhanced understanding of flow structures and their stability. The advantages of these techniques are shown for a range of flow situations, most of which are turbulent. It is shown that by these methods additional insight into complex flow situations can be gained. Such insight has been found to be needed for the flow in straight and 90 degree curved pipes. The so-called swirl switching phenomenon is investigated, which is a large scale oscillation of the flow after the bend. This phenomenon is classified into a low frequency and a high frequency switching, each with its own mechanism of formation. It is shown that while the low frequency switching stems from very-large-scale motions created in the upstream pipe, the high frequency switching results from the bend itself, making it an inherent property of the system. The second set of studies consider swirling flow in combustor-related geometries, using both high and low swirl levels. These investigations show highly energetic unsteady structures in the strongly vortical regions. The spatial symmetry of these flow modes reflect the level of confinement. While the vortices that are weakly confined show unsteady modes reflecting their displacement, the strongly confined vortices show low-order multipole deformations. For the low swirl burner, which is the only reacting flow considered, the flame is stabilized without the presence of vortex breakdown. To be able to investigate how the flame is anchored above the burner, an extended version of DMD (EDMD) is introduced, which helps to couple the flow with the flame. Using this method, a mechanism contributing to the flame stabilization is isolated. The third and final set of studies involve flow around cylinders and beams. These objects are flexible and respond to the forces that the flow exerts on them. For the flow around cylinders, which are connected to a spring system, the natural frequency of the spring-cylinder system and the frequency from the von Karman vortex shedding are the two a priori known frequencies of the system. Three different flow regimes are considered, one where the two frequencies are similar, giving resonance, and two cases where one frequency is far above/below the other. For flow around a single cylinder, an unexpected high energy low frequency mode is found off-resonance, which is argued to contribute greatly to the chaotic behaviour for the case with the loose spring. For a multiple cylinder array, while the strong low frequency mode found for the single cylinder case has been suppressed, an unexpected synchronization is seen. Considering the flow around a stiff and a flexible beam, a strong beat frequency is found for the lift force. While the beating is seen to be regular for the flexible beam, it appears intermittently for the stiff beam. The flow behaviour giving rise to this forcing is elucidated using the POD and DMD analyses