Strongly Monotone Drawings of Planar Graphs

A straight-line drawing of a graph is a monotone drawing if for each pair of vertices there is a path which is monotonically increasing in some direction, and it is called a strongly monotone drawing if the direction of monotonicity is given by the direction of the line segment connecting the two vertices. We present algorithms to compute crossing-free strongly monotone drawings for some classes of planar graphs; namely, 3-connected planar graphs, outerplanar graphs, and 2-trees. The drawings of 3-connected planar graphs are based on primal-dual circle packings. Our drawings of outerplanar graphs are based on a new algorithm that constructs strongly monotone drawings of trees which are also convex. For irreducible trees, these drawings are strictly convex

A duality transform for realizing convex polytopes with small integer coordinates

Wir entwickeln eine Dualitätstransformation für Polyeder, die eine Einbettung auf dem polynomiellen Gitter berechnet, wenn das ursprüngliche Polyeder auf einem polynomiellen Gitter gegeben ist. Die Konstruktion erfordert einen beschränkten Knotengrad des Polytop-Graphen, funktioniert aber im allgemeinen Fall für die Klasse der Stapelpolytope. Als Konsequenz können wir zeigen, dass sich die "Truncated Polytopes" auf einem polynomiellen Gitter realisieren lassen. Dieses Ergebnis gilt für jede (feste) Dimension.We study realizations of convex polytopes with small integer coordinates. We develop an efficient duality transform, that allows us to go from an efficient realization of a convex polytope to an efficient realization of its dual.Our methods prove to be especially efficient for realizing the class of polytopes dual to stacked polytopes, known as truncated polytopes. We show that every 3d truncated polytope with n vertices can be realized on an integer grid of size O(n^(9lg(6)+1)), and in R^d the required grid size is n^(O(d^2*lg(d))). The class of truncated polytopes is only the second nontrivial class of polytopes, the first being the class of stacked polytopes, for which realizations on a polynomial size integer grid are known to exists

Drawing Planar Cubic 3-Connected Graphs with Few Segments: Algorithms and Experiments

A drawing of a graph can be understood as an arrangement of geometric objects. In the most natural setting the arrangement is formed by straight-line segments. Every cubic planar 3-connected graph with n vertices has such a drawing with only n/2 +3 segments, matching the lower bound. This result is due to Mondal et al. [J. of Comb. Opt., 25], who gave an algorithm for constructing such drawings. We introduce two new algorithms that also produce drawings with n/2+3 segments. One algorithm is based on a sequence of dual edge contractions, the other is based on a recursion of nested cycles. We also show a flaw in the algorithm of Mondal et al. and present a fix for it. We then compare the performance of these three algorithms by measuring angular resolution, edge length and face aspect ratio of the constructed drawings. We observe that the corrected algorithm of Mondal et al. mostly outperforms the other algorithms, especially in terms of angular resolution. However, the new algorithms perform better in terms of edge length and minimal face aspect ratio

Organic Acids: The Pools of Fixed Carbon Involved in Redox Regulation and Energy Balance in Higher Plants

Organic acids are synthesized in plants as a result of the incomplete oxidation of photosynthetic products and represent the stored pools of fixed carbon accumulated due to different transient times of conversion of carbon compounds in metabolic pathways. When redox level in the cell increases, e.g., in conditions of active photosynthesis, the tricarboxylic acid (TCA) cycle in mitochondria is transformed to a partial cycle supplying citrate for the synthesis of 2-oxoglutarate and glutamate (citrate valve), while malate is accumulated and participates in the redox balance in different cell compartments (via malate valve). This results in malate and citrate frequently being the most accumulated acids in plants. However, the intensity of reactions linked to the conversion of these compounds can cause preferential accumulation of other organic acids, e.g., fumarate or isocitrate, in higher concentrations than malate and citrate. The secondary reactions, associated with the central metabolic pathways, in particularly with the TCA cycle, result in accumulation of other organic acids that are derived from the intermediates of the cycle. They form the additional pools of fixed carbon and stabilize the TCA cycle. Trans-aconitate is formed from citrate or cis-aconitate, accumulation of hydroxycitrate can be linked to metabolism of 2-oxoglutarate, while 4-hydroxy-2-oxoglutarate can be formed from pyruvate and glyoxylate. Glyoxylate, a product of either glycolate oxidase or isocitrate lyase, can be converted to oxalate. Malonate is accumulated at high concentrations in legume plants. Organic acids play a role in plants in providing redox equilibrium, supporting ionic gradients on membranes, and acidification of the extracellular medium

Drawing planar cubic 3-connected graphs with few segments: algorithms & experiments

A drawing of a graph can be understood as an arrangement of geometric objects. In the most natural setting the arrangement is formed by straight-line segments. Every cubic planar 3-connected graph with n\u3cbr/\u3en\u3cbr/\u3e vertices has such a drawing with only n/2+3\u3cbr/\u3en/2+3\u3cbr/\u3e segments, matching the lower bound. This result is due to Mondal et al. [J. of Comb. Opt., 25], who gave an algorithm for constructing such drawings. We introduce two new algorithms that also produce drawings with n/2+3\u3cbr/\u3en/2+3\u3cbr/\u3e segments. One algorithm is based on a sequence of dual edge contractions, the other is based on a recursion of nested cycles. We also show a flaw in the algorithm of Mondal et al. and present a fix for it. We then compare the performance of these three algorithms by measuring angular resolution, edge length and face aspect ratio of the constructed drawings. We observe that the corrected algorithm of Mondal et al. mostly outperforms the other algorithms, especially in terms of angular resolution. However, the new algorithms perform better in terms of edge length and minimal face aspect ratio