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

The Grid Sketcher: An AutoCad-based tool for conceptual design processes

Sketching with pencil and paper is reminiscent of the varied, rich, and loosely defined formal processes associated with conceptual design. Architects actively engage such creative paradigms in their exploration and development of conceptual design solutions. The Grid Sketcher, as a conceptual sketching tool, presents one possible computer implementation for enhancing and supporting these processes. It effectively demonstrates the facility with which current technology and the computing environment can enhance and simulate sketching intents and expectations; Typically with respect to design, the position taken is that the two are virtually void of any fundamental commonality. A designer\u27s thoughts are intuitive, at times irrational, and rarely follow consistently identifiable patterns. Conversely, computing requires predictability in just these endeavors. The computing environment, as commonly defined, can not reasonably expect to mimic the typically human domain of creative design. In this context, this thesis accentuates the computer\u27s role as a form generator as opposed to a form evaluator. The computer, under the influence of certain contextual parameters can, however, provide the designer with a rich and elegant set of forms that respond through algorithmics to the designer\u27s creative intents. (Abstract shortened by UMI.)

A New Tool for Rectangular Dualization

OcORD is a software tool for rectangular dualization. Rectangular dualization is a dual representation of a plane graph introduced in the early seventies. It proved to be effective in applications such as architectural space planning and VLSI floorplanning. However, not all plane graphs admit a rectangular dual, which imposes severe limitations on its use in other applications. OcORD aims at freeing rectangular dualization from such restrictions and proving its effectiveness in graph visualization. This is achieved in two ways. Firstly, OcORD features a new linear-time algorithm creating a rectangular dual of any plane graph. Secondly, it shows how nice drawings of a graph can be easily obtained from its rectangular dual. Finally, the automatic generation of a Virtual World through rectangular dualization is described. [DOI: 10.1685/CSC09301] About DO

Efficient Generation of Rectangulations via Permutation Languages

A generic rectangulation is a partition of a rectangle into finitely many interior-disjoint rectangles, such that no four rectangles meet in a point. In this work we present a versatile algorithmic framework for exhaustively generating a large variety of different classes of generic rectangulations. Our algorithms work under very mild assumptions, and apply to a large number of rectangulation classes known from the literature, such as generic rectangulations, diagonal rectangulations, 1-sided/area-universal, block-aligned rectangulations, and their guillotine variants. They also apply to classes of rectangulations that are characterized by avoiding certain patterns, and in this work we initiate a systematic investigation of pattern avoidance in rectangulations. Our generation algorithms are efficient, in some cases even loopless or constant amortized time, i.e., each new rectangulation is generated in constant time in the worst case or on average, respectively. Moreover, the Gray codes we obtain are cyclic, and sometimes provably optimal, in the sense that they correspond to a Hamilton cycle on the skeleton of an underlying polytope. These results are obtained by encoding rectangulations as permutations, and by applying our recently developed permutation language framework

Area-Universal Rectangular Layouts

A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. Rectangular layouts appear in various applications: as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it might hence be desirable if one rectangular layout can represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. More generally, given any rectangular layout L and any assignment of areas to its regions, we show that there can be at most one layout (up to horizontal and vertical scaling) which is combinatorially equivalent to L and achieves a given area assignment. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists.Comment: 19 pages, 16 figure

Transversal structures on triangulations: a combinatorial study and straight-line drawings

This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edge-labelling and consists of two bipolar orientations that are transversal. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straight-line drawing algorithm for irreducible triangulations. For a random irreducible triangulation with $n$ vertices, the grid size of the drawing is asymptotically with high probability $11n/27\times 11n/27$ up to an additive error of \cO(\sqrt{n}). In contrast, the best previously known algorithm for these triangulations only guarantees a grid size $(\lceil n/2\rceil -1)\times \lfloor n/2\rfloor$.Comment: 42 pages, the second version is shorter, focusing on the bijection (with application to counting) and on the graph drawing algorithm. The title has been slightly change