Strictly convex drawings of planar graphs

Every three-connected planar graph with n vertices has a drawing on an O(n^2) x O(n^2) grid in which all faces are strictly convex polygons. These drawings are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids. More generally, a strictly convex drawing exists on a grid of size O(W) x O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds are obtained when the faces have fewer sides. In the proof, we derive an explicit lower bound on the number of primitive vectors in a triangle.Comment: 20 pages, 13 figures. to be published in Documenta Mathematica. The revision includes numerous small additions, corrections, and improvements, in particular: - a discussion of the constants in the O-notation, after the statement of thm.1. - a different set-up and clarification of the case distinction for Lemma

Smart Solutions: Smart Grid Demokit

Treball desenvolupat dins el marc del programa 'European Project Semester'.The purpose of this report is to justify the design choices of the smart grid demo kit. Something had to be designed to make a smart grid clear for people who have little knowledge about smart grids. The product had to be appealing and clear for people to understand. And eventually should be usable, for example, on an information market. The first part of the research consisted of looking how to shape the whole system. How the 'tiles' had to look to be interactive for users and what they should feature. One part of this was doing research to get to know more about the already existing knowledge amount users. Another research investigated what appeals the most to the users. After this, a concept was created in compliance with the group and the client. The concept consists of hexagonal tiles, each with a different function: houses, solar panels, wind turbines, factories and energy storages. These tiles are all different parts of a smart grid. When combining these tiles, it can be made clear to users how smart grids work. The tiles are fabricated using a combination of 3D printing and laser cutting. The tiles have laser cut symbols on top of them to show what part of the smart grid they are. Digital LED strips are on top of the tiles to show the direction of the energy flow, and the colors indicate if the tile is producing or consuming power from the grid. The tiles are connected to each other by the so called “grid blocks”. These blocks make up the central power grid and are also lighting up by LED strips. Each tile is equipped with a microcontroller which controls the LED strips and makes it possible for the different tiles to “talk” with each other. Using this, the central tile knows which tiles are connected to the system. The central tile controls all tiles and runs the simulation of the smart grid. For further development of the project, it can be investigated how to control and adjust the system from an external system, for example by a tablet. The final product consists of five tiles connected by seven grid blocks which show how a smart grid works

Shortest path embeddings of graphs on surfaces

The classical theorem of F\'{a}ry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of F\'{a}ry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces. We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property. Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil-Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of reviewer