Bidimensionality of Geometric Intersection Graphs

Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric intersection graphs GB where each body of the collection B is represented by a vertex, and two vertices of GB are adjacent if the intersection of the corresponding bodies is non-empty. For such graph classes and under natural restrictions on their maximum degree or subgraph exclusion, we prove that the relation between their treewidth and the maximum size of a grid minor is linear. These combinatorial results vastly extend the applicability of all the meta-algorithmic results of the bidimensionality theory to geometrically defined graph classes

Shapes of interacting RNA complexes

Shapes of interacting RNA complexes are studied using a filtration via their topological genus. A shape of an RNA complex is obtained by (iteratively) collapsing stacks and eliminating hairpin loops. This shape-projection preserves the topological core of the RNA complex and for fixed topological genus there are only finitely many such shapes.Our main result is a new bijection that relates the shapes of RNA complexes with shapes of RNA structures.This allows to compute the shape polynomial of RNA complexes via the shape polynomial of RNA structures. We furthermore present a linear time uniform sampling algorithm for shapes of RNA complexes of fixed topological genus.Comment: 38 pages 24 figure

The Dara Building (Grote Koppel), Amersfoort, Netherlands

The Dara Building in Amersfoort, Netherlands, makes a significant contribution to FAT Architecture’s on-going research into the creative potential of historical reference and repetition, in combination with digital and prefabricated construction techniques to generate new meanings in architecture. Griffiths was the lead architect on the project. Its design responded to a number of questions: How can a modern building integrate with and extend the meanings of an historic context? How can differentiation and variety be achieved using repetition? How can precast concrete construction be used to create expressive popular iconography and communicate cultural values about architecture? Can an art-based architectural practice be successful in a market driven environment? Its methodology included numerous site visits to understand the site’s complexity and latent potential, discussions with local planning authorities to get a sense of the Dutch legislation and regulations for historic contexts and typological research, drawing on the traditions of baroque influenced, gable fronted Dutch architecture. A variety of programmatic solutions, spatial permutations, and the three-dimensional complexity of the building and its surroundings were tested through extensive physical model making and other forms of digital visualisation. The innovative external wall and window panels of the building were generated by drawing and re-drawing, then interpreting these design motifs in digital format, which were then transferred directly to Dutch prefabricated concrete manufacturer, Hibex. The architects then collaborated closely with the manufacturer to produce the building’s signature prefabricated façade panels. The building has been favourably reviewed in the architectural media, including in Building Design, Blue Print and Domus. It is regularly featured in lectures and exhibitions about the work of FAT delivered nationally and internationally including at London Metropolitan University in 2009, the Walker Art Center in Minneapolis in 2009 and the Strelka Institute in Moscow in 2010

Decomposition of multiple packings with subquadratic union complexity

Suppose $k$ is a positive integer and $\mathcal{X}$ is a $k$-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most $k$ sets. Suppose there is a function $f(n)=o(n^2)$ with the property that any $n$ members of $\mathcal{X}$ determine at most $f(n)$ holes, which means that the complement of their union has at most $f(n)$ bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that $\mathcal{X}$ can be decomposed into at most $p$ ($1$-fold) packings, where $p$ is a constant depending only on $k$ and $f$.Comment: Small generalization of the main result, improvements in the proofs, minor correction

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