Drawing Graphs as Spanners

We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $\Gamma$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$, the ratio between the minimum length of any path from $u$ to $v$ and the Euclidean distance between $u$ and $v$ is small. The maximum such ratio, over all pairs of vertices of $G$, is the spanning ratio of $\Gamma$. First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio $1$, a proper straight-line drawing with spanning ratio $1$, and a planar straight-line drawing with spanning ratio $1$ are NP-complete, $\exists \mathbb R$-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a vertex. Second, we show that moving from spanning ratio $1$ to spanning ratio $1+\epsilon$ allows us to draw every graph. Namely, we prove that, for every $\epsilon>0$, every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than $1+\epsilon$. Third, our drawings with spanning ratio smaller than $1+\epsilon$ have large edge-length ratio, that is, the ratio between the length of the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line drawing with constant spanning ratio

Archaeological perspectives for northern Patagonia: Cueva Huenul 1 Site (Neuquen Province, Argentina)

Northern Neuquén Province (Pehuenches Dept., Argentina) is barely known from an archaeological perspective, though it is centrally placed in terms of several large-scale key issues in the peopling of South America: the extinction of the megafauna and its causes, early human presence, and the existence of archaeological discontinuities during the Mid-Holocene. In this paper we present the first body of paleoecological and archaeological data for Cueva Huenul 1 site, recently excavated, which offers a sedimentary sequence extending during the last of 16.000 calendar years. Initially, we present a chrono-stratigraphic frame for the site, including new tephro-chronological information. On this basis, four temporal components are defined, providing the historical scheme for the analysis of the recovered evidences that include: archaeofaunas (paleontological and archaeological), archaeobotany, lithic and ceramic technology, and rockart. These results at a site scale provide a first approach to a discussion of macro-regional processes, as well as the basis for the continuation of our research.El norte de la provincia de Neuquén (Depto. Pehuenches, Argentina) es muy poco conocido a nivel arqueológico, a pesar de estar ubicado en una posición central en relación con distintos temas clave del poblamiento humano de Sudamérica, incluyendo la extinción de la megafauna y sus causas, el poblamiento humano inicial y la existencia de discontinuidades arqueológicas en el Holoceno medio. En este trabajo se presenta el primer cuerpo de resultados paleoecológicos y arqueológicos para el sitio Cueva Huenul 1, recientemente excavado, que ofrece una secuencia sedimentaria que se extiende durante los últimos 16.000 años calendáricos. Estas evidencias incluyen el desarrollo de un marco crono-estratigráfico para el sitio, que aporta novedosa información tefro-cronológica. A partir de este análisis, se definen cuatro componentes temporales, sobre los cuales se asienta el estudio de los materiales recuperados: evidencias faunísticas (paleontológicas y arqueológicas), arqueobotánicas, líticas, cerámicas y de arte rupestre. Estos resultados en escala de sitio proveen una primera instancia de evaluación de procesos en escala macro-regional, así como las bases para la continuación de este proyecto.Facultad de Ciencias Naturales y Muse

On the planar edge-length ratio of planar graphs

The edge-length ratio of a straight-line drawing of a graph is the ratio between the lengths of the longest and of the shortest edge in the drawing. The planar edge-length ratio of a planar graph is the minimum edge-length ratio of any planar straight-line drawing of the graph. In this paper, we study the planar edge-length ratio of planar graphs. We prove that there exist n-vertex planar graphs whose planar edge-length ratio is in Ω(n); this bound is tight. We also prove upper bounds on the planar edge-length ratio of several families of planar graphs, including series-parallel graphs and bipartite planar graphs

On the planar edge-length ratio of planar graphs

The edge-length ratio of a straight-line drawing of a graph is the ratio between the lengths of the longest and of the shortest edge in the drawing. The planar edge-length ratio of a planar graph is the minimum edge-length ratio of any planar straight-line drawing of the graph.In this paper, we study the planar edge-length ratio of planar graphs. We prove that there exist $n$-vertex planar graphs whose planar edge-length ratio is in $\Omega(n)$; this bound is tight. We also prove upper bounds on the planar edge-length ratio of several families of planar graphs, including series-parallel graphs and bipartite planar graphs