Drawing Planar Graphs with Few Geometric Primitives

We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let $n$ denote the number of vertices of a graph. We show that trees can be drawn with $3n/4$ straight-line segments on a polynomial grid, and with $n/2$ straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with $(8n-17)/3$ segments on an $O(n)\times O(n^2)$ grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with $3n/2$ edges on an $O(n)\times O(n^2)$ grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only $(5n - 11)/3$ arcs. This is significantly smaller than the lower bound of $2n$ for line segments for a nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2017

Beyond Outerplanarity

We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., convex drawings. We consider two families of graph classes with nice convex drawings: outer $k$-planar graphs, where each edge is crossed by at most $k$ other edges; and, outer $k$-quasi-planar graphs where no $k$ edges can mutually cross. We show that the outer $k$-planar graphs are $(\lfloor\sqrt{4k+1}\rfloor+1)$-degenerate, and consequently that every outer $k$-planar graph can be $(\lfloor\sqrt{4k+1}\rfloor+2)$-colored, and this bound is tight. We further show that every outer $k$-planar graph has a balanced separator of size $O(k)$. This implies that every outer $k$-planar graph has treewidth $O(k)$. For fixed $k$, these small balanced separators allow us to obtain a simple quasi-polynomial time algorithm to test whether a given graph is outer $k$-planar, i.e., none of these recognition problems are NP-complete unless ETH fails. For the outer $k$-quasi-planar graphs we prove that, unlike other beyond-planar graph classes, every edge-maximal $n$-vertex outer $k$-quasi planar graph has the same number of edges, namely $2(k-1)n - \binom{2k-1}{2}$. We also construct planar 3-trees that are not outer $3$-quasi-planar. Finally, we restrict outer $k$-planar and outer $k$-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence on the boundary is a cycle in the graph. For each $k$, we express closed outer $k$-planarity and \emph{closed outer $k$-quasi-planarity} in extended monadic second-order logic. Thus, closed outer $k$-planarity is linear-time testable by Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Lombardi Drawings of Graphs

We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure

Efecto promotor del agregado de rodio a catalizadores de Co/ZnAl2O4 para la combustión catalítica de hidrocarburos

Una serie de catalizadores de cobalto soportados sobre aluminato de zinc (Cox-ZnAl) modificados con Rh, fue preparada, caracterizada mediante diversas técnicas (BET, XRD, TPR, DRS, FTIR y XPS) y aplicados en las reacciones de oxidación catalítica de propano y naftaleno

Stomach contents of sperm whales Physeter macrocephalus stranded in the North Sea 1990-1996

Stomach contents of 17 sperm whales Physeter macrocephalus stranded in Scotland and Denmark during 1990-96 were analysed. All were sub-adult or adult males and stranded between November and March. They had presumably entered the North Sea during their southward migration from feeding grounds in Arctic waters. Other studies indicate that the majority of the whales were apparently healthy. The diet of these whales was found to consist almost entirely of cephalopods, principally squid of the genus Gonatus (hereafter 'Gonatus', but probably G. fabricii, an oceanic species characteristic of Arctic waters). The other prey species identified were also mostly oceanic cephalopods: the squids Histioteuthis bonnellii, Teuthowenia megalops and Todarodes sagittatus and the octopus Haliphron atlanticus. Although these results are consistent with other recent studies in the area based on single stranded whales, they differ from results of work on whales caught during commercial whaling operations in Icelandic waters (1960s to 1980s) in that little evidence of predation on fish was found in the present study. Remains of single individuals of the veined squid Loligo forbesi, the northern octopus Eledone cirrhosa and the saithe Pollachius virens provided the only possible evidence of feeding in the North Sea. We infer that sperm whales do not enter the North Sea to feed. The timing, and large and uniform sizes of the Gonatus species eaten (most had mantle lengths in the range 195 to 245 mm), as estimated from measurements of the lower beaks, and the seasonality of the strandings is consistent with the whales having fed on mature squid, possibly spawning concentrations--as has recently been reported for bottlenose whales. Assuming that the diet recorded in this study was representative of sperm whales during the feeding season, as much as 500000 t of Gonatus could be removed by sperm whales in Norwegian waters each year and up to 3 times that figure from the eastern North Atlantic as a whole. Evidence from other studies indicates that Gonatus is an important food resource for a wide range of marine predators in Arctic waters

Efficient Enumeration of Bipartite Subgraphs in Graphs

Subgraph enumeration problems ask to output all subgraphs of an input graph that belongs to the specified graph class or satisfy the given constraint. These problems have been widely studied in theoretical computer science. As far, many efficient enumeration algorithms for the fundamental substructures such as spanning trees, cycles, and paths, have been developed. This paper addresses the enumeration problem of bipartite subgraphs. Even though bipartite graphs are quite fundamental and have numerous applications in both theory and application, its enumeration algorithms have not been intensively studied, to the best of our knowledge. We propose the first non-trivial algorithms for enumerating all bipartite subgraphs in a given graph. As the main results, we develop two efficient algorithms: the one enumerates all bipartite induced subgraphs of a graph with degeneracy $k$ in $O(k)$ time per solution. The other enumerates all bipartite subgraphs in $O(1)$ time per solution

Un proceso en cascada para la obtención de y-valerolactona a partir de fructosa

En el presente trabajo se logró deshidratar fructosa para obtener 5- hidroximetil furfural (HMF), molécula plataforma que permite una posterior obtención de biocombustibles y polímeros. La reacción de deshidratación se realizó usando como catalizadores líquidos iónicos (LIs) preparados en nuestro laboratorio a 80°C durante 1 h.Centro de Investigación y Desarrollo en Ciencias Aplicada