Fun with Fonts: Algorithmic Typography

Over the past decade, we have designed six typefaces based on mathematical theorems and open problems, specifically computational geometry. These typefaces expose the general public in a unique way to intriguing results and hard problems in hinged dissections, geometric tours, origami design, computer-aided glass design, physical simulation, and protein folding. In particular, most of these typefaces include puzzle fonts, where reading the intended message requires solving a series of puzzles which illustrate the challenge of the underlying algorithmic problem.Comment: 14 pages, 12 figures. Revised paper with new glass cane font. Original version in Proceedings of the 7th International Conference on Fun with Algorithm

Cubic Augmentation of Planar Graphs

In this paper we study the problem of augmenting a planar graph such that it becomes 3-regular and remains planar. We show that it is NP-hard to decide whether such an augmentation exists. On the other hand, we give an efficient algorithm for the variant of the problem where the input graph has a fixed planar (topological) embedding that has to be preserved by the augmentation. We further generalize this algorithm to test efficiently whether a 3-regular planar augmentation exists that additionally makes the input graph connected or biconnected. If the input graph should become even triconnected, we show that the existence of a 3-regular planar augmentation is again NP-hard to decide.Comment: accepted at ISAAC 201

Optimal Time-Convex Hull under the Lp Metrics

We consider the problem of computing the time-convex hull of a point set under the general $L_p$ metric in the presence of a straight-line highway in the plane. The traveling speed along the highway is assumed to be faster than that off the highway, and the shortest time-path between a distant pair may involve traveling along the highway. The time-convex hull ${TCH}(P)$ of a point set $P$ is the smallest set containing both $P$ and \emph{all} shortest time-paths between any two points in ${TCH}(P)$. In this paper we give an algorithm that computes the time-convex hull under the $L_p$ metric in optimal $O(n\log n)$ time for a given set of $n$ points and a real number $p$ with $1\le p \le \infty$

Searching for equilibrium positions in a game of political competition with restrictions

This paper considers a problem of political economy in which a Nash equilibrium study is performed in a proposed game with restrictions where the two major parties in a country vary their position within a politically flexible framework to increase their number of voters. The model as presented fits the reality of many countries. Moreover, it avoids the uniqueness of equilibrium positions. The problem is stated and solved from a geometric point of view

La regeneración del pinsapar en la sierra de Grazalema. I: análisis de la fase de plántula

Se estudia la dinámica de la regeneración del pinsapar en la Sierra de Grazalema. Previamente se ha llevado a cabo una diferenciación del conjunto del pinsapar a partir de la información disponible de inventarios, fotointerpretación, bibliografía y datos de campo, dando como resultados cuatro tipos estructurales: Pinsapar puro estructurado, Pinsapar puro latizal-fustal, Quejigal con pinsapos y Encinar con pinsapos. En este estudio, el análisis del proceso de regeneración se centra en el regenerado en estado de plántula, donde la mortalidad puede tener una incidencia enorme debida especialmente a la sequía estival propia del ámbito mediterráneo. Se ha establecido un dispositivo de muestreo sistemático en los cuatro estratos, medido una vez finalizada la germinación de semillas del año y después del verano. Con los datos obtenidos se ha estudiado tanto la incorporación, como la mortalidad de nuevas plántulas. Se han encontrado diferencias significativas entre los estratos de pinsapar puro y los mixtos, tanto en incorporación de nuevas plántulas como en mortalidad. También existen diferencias dentro de los dos estratos de pinsapar pur

La regeneración del pinsapar en la sierra de Grazalema. II: estructura y dinámica del regenerado consolidado en el pinsapar puro

En los tipos estructurales de pinsapar puro, se estudia la dinámica y la estructura del regenerado consolidado (plantas de más de un año de edad que no alcanzan las dimensiones de los pies menores, es decir, menos de 1,5 m de altura). El motivo de separar este grupo del de plántulas menores de un año, estriba en las diferentes posibilidades de supervivencia de uno y otro estado ya que en el regenerado consolidado va a depender en mayor medida del grado de competencia al que se vea sometido que de las condiciones locales del medio en que se asienta. El análisis de la distribución de alturas por clases muestra importantes diferencias significativas entre ambos estratos de pinsapar puro, para la clase < 15 cm pero no para las demás clases. Se ha analizado la evolución de la densidad del regenerado por clases de altura desde la fase de plántula hasta la de pie menor, obteniendo una gráfica descendente similar en ambos estratos con una caída brusca inexplicada en la clase de 100 a 150 cm, seguida de una subida en la clase de pies menores atribuida al estancamiento en el crecimiento en altura. El estudio de la relación altura edad demuestra que ésta es significativa, para el regenerado, pero no para los pies menores, indicando así que este grupo se compone de árboles dominados de numerosas y muy distintas edades. El regenerado consolidado constituye una reserva de regeneración a la espera, cuyos individuos aprovecharían las perturbaciones en la estructura para prosperar. Puede decirse que en las circunstancias actuales y frente a perturbaciones no catastróficas, la persistencia del pinsapar está asegurad

Flip Graphs of Degree-Bounded (Pseudo-)Triangulations

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant $k$. In particular, we consider triangulations of sets of $n$ points in convex position in the plane and prove that their flip graph is connected if and only if $k > 6$; the diameter of the flip graph is $O(n^2)$. We also show that, for general point sets, flip graphs of pointed pseudo-triangulations can be disconnected for $k \leq 9$, and flip graphs of triangulations can be disconnected for any $k$. Additionally, we consider a relaxed version of the original problem. We allow the violation of the degree bound $k$ by a small constant. Any two triangulations with maximum degree at most $k$ of a convex point set are connected in the flip graph by a path of length $O(n \log n)$, where every intermediate triangulation has maximum degree at most $k+4$.Comment: 13 pages, 12 figures, acknowledgments update

A class of solvable Lie algebras and their Casimir Invariants

A nilpotent Lie algebra n_{n,1} with an (n-1) dimensional Abelian ideal is studied. All indecomposable solvable Lie algebras with n_{n,1} as their nilradical are obtained. Their dimension is at most n+2. The generalized Casimir invariants of n_{n,1} and of its solvable extensions are calculated. For n=4 these algebras figure in the Petrov classification of Einstein spaces. For larger values of n they can be used in a more general classification of Riemannian manifolds.Comment: 16 page

Planar subgraphs without low-degree nodes

We study the following problem: given a geometric graph G and an integer k, determine if G has a planar spanning subgraph (with the original embedding and straight-line edges) such that all nodes have degree at least k. If G is a unit disk graph, the problem is trivial to solve for k = 1. We show that even the slightest deviation from the trivial case (e.g., quasi unit disk graphs or k = 1) leads to NP-hard problems.Peer reviewe