Extending Orthogonal Planar Graph Drawings Is Fixed-Parameter Tractable

The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for fundamental representations such as planar and beyond-planar topological drawings. In this paper, we consider the extension problem for bend-minimal orthogonal drawings of planar graphs, which is among the most fundamental geometric graph drawing representations. While the problem was known to be NP-hard, it is natural to consider the case where only a small part of the graph is still to be drawn. Here, we establish the fixed-parameter tractability of the problem when parameterized by the size of the missing subgraph. Our algorithm is based on multiple novel ingredients which intertwine geometric and combinatorial arguments. These include the identification of a new graph representation of bend-equivalent regions for vertex placement in the plane, establishing a bound on the treewidth of this auxiliary graph, and a global point-grid that allows us to discretize the possible placement of bends and vertices into locally bounded subgrids for each of the above regions

Pop-tech-flat-fab

This paper for the EAAE / ARCC 2008 addresses the theme of simultaneity between the digital and analogue by examining the production of two projects. These are: a pair of prototype bus stops built in Sioux City1 and a shade structure for downtown Phoenix in the USA. The conceptual basis for both these projects coincides with the question of how "phenomenon attached to a certain locality”2 might be created through advanced methods of digital fabrication. Both projects offer an apology for rapid prototyping techniques applied to an understanding of "contextualism”3. Both projects are presented first as a contextual and symbolic response to an interpretation of "locality” and then re-appraised in technical terms. In both projects these technical aspects aim to advance not only the methods of physical production but also the transition of design methods to 1:1 fabrication. In the case of the Sioux City Bus Stops this idea is represented through an analysis of two-dimensional cutting techniques and developable surfaces. In the case of the Phoenix Shade project this idea is then developed through fully associative digital models. Together these projects attempt to accelerate the physical production of their symbolic and contextual content through a discussion on parametric modeling that allows an efficient production of a set of different permutations. By associating the symbolic/contextual with the parametric these projects suggest and alternative procedure to the traditional and prevalent trope of "digital architecture” and its co-dependence upon explicitly biomorphic, computational and quasinaturalistic language

On a Tree and a Path with no Geometric Simultaneous Embedding

Two graphs $G_1=(V,E_1)$ and $G_2=(V,E_2)$ admit a geometric simultaneous embedding if there exists a set of points P and a bijection M: P -> V that induce planar straight-line embeddings both for $G_1$ and for $G_2$. While it is known that two caterpillars always admit a geometric simultaneous embedding and that two trees not always admit one, the question about a tree and a path is still open and is often regarded as the most prominent open problem in this area. We answer this question in the negative by providing a counterexample. Additionally, since the counterexample uses disjoint edge sets for the two graphs, we also negatively answer another open question, that is, whether it is possible to simultaneously embed two edge-disjoint trees. As a final result, we study the same problem when some constraints on the tree are imposed. Namely, we show that a tree of depth 2 and a path always admit a geometric simultaneous embedding. In fact, such a strong constraint is not so far from closing the gap with the instances not admitting any solution, as the tree used in our counterexample has depth 4.Comment: 42 pages, 33 figure