The term biological motion has been coined by G. Johansson (1973) to refer to the ambulatory patterns of terrestrial bipeds and quadripeds. In this paper a computational theory of the visual perception of biological motion is proposed. The specific problem addressed is how the three dimensional structure and motions of animal limbs may be computed from the two dimensional motions of their projected images. It is noted that the limbs of animals typically do not move arbitrarily during ambulation. Rather, for anatomical reasons, they typically move in single planes for extended periods of time. This simple anatomical constraint is exploited as the basis for utilizing a "planarity assumption" in the interpretation of biological motion. The analysis proposed is: (1) divide the image into groups of two or three elements each; (2) test each group for pairwise-rigid planar motion; (3) combine the results from (2). Fundamental to the analysis are two 'structure from planar motion' propositions. The first states that the structure and motion of two points rigidly linked and rotating in a plane is recoverable from three orthographic projections. The second states that the structure and motion of three points forming two hinged rods constrained to move in a plane is recoverable from two orthographic projections. The psychological relevance of the analysis and possible interactions with top down recognition processes are discussed.

In the game of Kal-toh depicted in the television series Star Trek: Voyager, players attempt to create polyhedra by adding to a jumbled collection of metal rods. Inspired by this fictional game, we formulate graph-theoretical questions about polyhedral (triconnected and planar) subgraphs in an on-line environment. The problem of determining the existence of a polyhedral subgraph within a graph G is shown to be NP-hard, and we also give some non-trivial upper bounds for the problem of determining the minimum number of edge additions necessary to guarantee the existence of a polyhedral subgraph in G. A two-player formulation of Kal-toh is also explored, in which the first player to form a target subgraph is declared the winner. We show a polynomial-time solution for simple cases of this game but conjecture that the general problem is NP-hard.

In the game of Kal-toh depicted in the television series Star Trek: Voyager, players attempt to create polyhedra by adding to a jumbled collection of metal rods. Inspired by this fictional game, we formulate graph-theoretical questions about polyhedral (triconnected and planar) subgraphs in an on-line environment. The problem of determining the existence of a polyhedral subgraph within a graph G is shown to be NP-hard, and we also give some non-trivial upper bounds for the problem of determining the minimum number of edge additions necessary to guarantee the existence of a polyhedral subgraph in G. A two-player formulation of Kal-toh is also explored, in which the first player to form a target subgraph is declared the winner. We show a polynomial-time solution for simple cases of this game but conjecture that the general problem is NP-hard.

Consider a graph G with vertex set V in which each of the n vertices is assigned a number from the set {1, ..., k} for some positive integer k. This assignment phi is a labeling if all k numbers are used. If phi does not assign adjacent vertices the same label, then phi partitions V into k levels. In a level drawing, the y-coordinate of each vertex matches its label and the edges are drawn strictly y-monotone. This leads to level drawings in the xy-plane where all vertices with label j lie along the line lj = {(x, j) : x in Reals} and where each edge crosses any of the k horizontal lines lj for j in [1..k] at most once. A graph with such a labeling forms a level graph and is level planar if it has a level drawing without crossings.We first consider the class of level trees that are level planar regardless of their labeling. We call such trees unlabeled level planar (ULP). We describe which trees are ULP and provide linear-time level planar drawing algorithms for any labeling. We characterize ULP trees in terms of two forbidden subdivisions so that any other tree must contain a subtree homeomorphic to one of these. We also provide linear-time recognition algorithms for ULP trees. We then extend this characterization to all ULP graphs with five additional forbidden subdivisions, and provide linear-time recogntion and drawing algorithms for any given labeling.

In this study, we used different ways to synthetize four-coordinate zinc(II) and nickel(II) complexes with optically active Schiff base offering N2S2 coordination, i.e. N,N'-Bis(2-thiobenzylidene)-l,2-(R,R)cycholhexyl- enediaminatozinc(II) (1) and N,N'-Bis(2-thiobenzylidene)-l,2-(R,R)cychol- hexylenediaminatonickel(II) (2). We obtained the crystal structure of 2 and the pyridine adduct of 1, N,N'-Bis(2-thiobenzylidene)-l,2-(R,R)cycholhexylenediaminopyridylzinc(II) (1¡DPy). Coordination geometry around Zn atom in 1¡DPy is a distorted trigonal bipyramid. These structures were compared with the N2O2 stuctural analogues reported in the literature. We hope these chiral complexes can make contribution to the enzyme model studies in the future.

Recently, Formal Concept Analysis has proven to be an efficient method for the analysis and representation of information. However, the possibility to visualize concept hierarchies is being affected by the difficulty of drawing attractive diagrams automatically. Reducing the number of edge crossings seems to increase the readability of those drawings. This dissertation concerns with a mandatory prerequisite of this constraint, namely the characterization and visual representation of planar lattices. The manifold existing approaches and algorithms are thereby considered under a different point of view. It is well known that exactly the planar lattices (or planar posets) possess an additional order ``from left to right''. Our aim in this work is to define left-relations and left-orders more precisely and to describe several aspects of planar lattices with their help. The three approaches employed structure the work in as many parts: Left-relations on lattices allow a more efficient consideration of conjugate orders since they are uniquely determined by the sorting of the meet-irreducibles. Additionally, the restriction on the meet-irreducibles enables us to achieve an intuitive description of standard contexts of planar lattices similar to the consecutive-one property. With the help of left-relations on diagrams, planar lattices can indeed be drawn without edge crossings in the plane. Thereby, lattice-theoretically found left-orders can be detected in the graphical representation again. Furthermore, we modify the left-right-numbering algorithm in order to obtain attribute-additive and plane drawings of planar lattices. Finally, we will consider left-relations on contexts. They turn out to be fairly similar structures to the Ferrers-graphs. Planar lattices can be characterized by a property of these graphs, namely the bipartiteness. We will constructively prove this result. Subsequently, we can design an efficient algorithm that finds all non-similar plane diagrams of a lattice. / Die Formale Begriffsanalyse hat sich in den letzten Jahren als effizientes Werkzeug zur Datenanalyse und -repräsentation bewährt. Die Möglichkeit der visuellen Darstellung von Begriffshierarchien wird allerdings durch die Schwierigkeit, ansprechende Diagramme automatisch generieren zu können, beeinträchtigt. Offenbar sind Diagramme mit möglichst wenig Kantenkreuzungen für den menschlichen Anwender leichter lesbar. Diese Arbeit beschäftigt sich mit mit einer diesem Kriterium zugrunde liegenden Vorleistung, nämlich der Charakterisierung und Darstellung planarer Verbände. Die schon existierenden vielfältigen Ansätze und Methoden werden dabei unter einem neuen Gesichtspunkt betrachtet. Bekannterweise besitzen genau die planaren Verbände (bzw. planare geordnete Mengen) eine zusätzliche Ordnung "von links nach rechts". Unser Ziel in dieser Arbeit ist es, solche Links-Relationen bzw. Links-Ordnungen genauer zu definieren und verschiedene Aspekte planarer Verbände mit ihrer Hilfe zu beschreiben. Die insgesamt drei auftretenden Sichtweisen gliedern die Arbeit in ebensoviele Teile: Links-Relationen auf Verbänden erlauben eine effizientere Behandlung konjugierter Ordnungen, da sie durch die Anordnung der Schnitt-Irreduziblen schon eindeutig festgelegt sind. Außerdem erlaubt die Beschränkung auf die Schnitt-Irreduziblen eine anschauliche Beschreibung von Standardkontexten planarer Verbände ähnlich der consecutive-one property. Mit Hilfe der Links-Relationen auf Diagrammen können planare Verbände tatsächlich eben gezeichnet werden. Dabei lassen sich verbandstheoretisch ermittelte Links-Ordnungen in der graphischen Darstellung wieder finden. Weiterhin geben wir in eine Modifikation des left-right-numbering an, mit der planare Verbände merkmaladditiv und eben gezeichnet werden können. Schließlich werden wir Links-Relationen auf Kontexten betrachten. Diese stellen sich als sehr ähnlich zu Ferrers-Graphen heraus. Planare Verbände lassen sich durch eine Eigenschaft dieser Graphen, nämlich die Bipartitheit, charakterisieren. Wir werden dieses Ergebnis konstruktiv beweisen und darauf aufbauend einen effizienten Algorithmus angeben, mit dem alle nicht-ähnlichen ebenen Diagramme eines Verbandes bestimmt werden können.

We can visualize a graph by producing a geometric representation of the graph in which each node is represented by a single point on the plane, and each edge is represented by a curve that connects its two endpoints. Directed graphs are often used to model hierarchical structures; in order to visualize the hierarchy represented by such a graph, it is desirable that a drawing of the graph reflects this hierarchy. This can be achieved by drawing all the edges in the graph such that they all point in an upwards direction. A graph that has a drawing in which all edges point in an upwards direction and in which no edges cross is known as an upward planar graph. Unfortunately, testing if a graph is upward planar is NP-complete. Parameterized complexity is a technique used to find efficient algorithms for hard problems, and in particular, NP-complete problems. The main idea is that the complexity of an algorithm can be constrained, for the most part, to a parameter that describes some aspect of the problem. If the parameter is fixed, the algorithm will run in polynomial time. In this thesis, we investigate contracting an edge in an upward planar graph that has a specified embedding, and show that we can determine whether or not the resulting embedding is upward planar given the orientation of the clockwise and counterclockwise neighbours of the given edge. Using this result, we then show that under certain conditions, we can join two upward planar graphs at a vertex and obtain a new upward planar graph. These two results expand on work done by Hutton and Lubiw. Finally, we show that a biconnected graph has at most <i>k</i>!8^<i>k</i>-1 planar embeddings, where <i>k</i> is the number of triconnected components. By using an algorithm by Bertolazzi et al. that tests whether a given embedding is upward planar, we obtain a parameterized algorithm, where the parameter is the number of triconnected components, for testing the upward planarity of a biconnected graph. This algorithm runs in <i>O</i>(<i>k</i>!8^<i>k</i><i>n</i>³) time.

We can visualize a graph by producing a geometric representation of the graph in which each node is represented by a single point on the plane, and each edge is represented by a curve that connects its two endpoints. Directed graphs are often used to model hierarchical structures; in order to visualize the hierarchy represented by such a graph, it is desirable that a drawing of the graph reflects this hierarchy. This can be achieved by drawing all the edges in the graph such that they all point in an upwards direction. A graph that has a drawing in which all edges point in an upwards direction and in which no edges cross is known as an upward planar graph. Unfortunately, testing if a graph is upward planar is NP-complete. Parameterized complexity is a technique used to find efficient algorithms for hard problems, and in particular, NP-complete problems. The main idea is that the complexity of an algorithm can be constrained, for the most part, to a parameter that describes some aspect of the problem. If the parameter is fixed, the algorithm will run in polynomial time. In this thesis, we investigate contracting an edge in an upward planar graph that has a specified embedding, and show that we can determine whether or not the resulting embedding is upward planar given the orientation of the clockwise and counterclockwise neighbours of the given edge. Using this result, we then show that under certain conditions, we can join two upward planar graphs at a vertex and obtain a new upward planar graph. These two results expand on work done by Hutton and Lubiw. Finally, we show that a biconnected graph has at most <i>k</i>!8^<i>k</i>-1 planar embeddings, where <i>k</i> is the number of triconnected components. By using an algorithm by Bertolazzi et al. that tests whether a given embedding is upward planar, we obtain a parameterized algorithm, where the parameter is the number of triconnected components, for testing the upward planarity of a biconnected graph. This algorithm runs in <i>O</i>(<i>k</i>!8^<i>k</i><i>n</i>³) time.

This dissertation has two principal components: the dimension of posets with planar cover graphs, and the cartesian product of posets whose cover graphs have hamiltonian cycles that parse into symmetric chains. Posets of height two can have arbitrarily large dimension. In 1981, Kelly provided an infinite sequence of planar posets that shows that the dimension of planar posets can also be arbitrarily large. However, the height of the posets in this sequence increases with the dimension. In 2009, Felsner, Li, and Trotter conjectured that for each integer h at least 2, there exists a least positive integer c(h) so that if P is a poset with a planar cover graph (the class of posets with planar cover graphs includes the class of planar posets) and the height of P is h, then the dimension of P is at most c(h). In the first principal component of this dissertation we prove this conjecture. We also give the best known lower bound for c(h), noting that this lower bound is far from the upper bound. In the second principal component, we consider posets with the Hamiltonian Cycle--Symmetric Chain Partition (HC-SCP) property. A poset of width w has this property if its cover graph has a hamiltonian cycle which parses into w symmetric chains. This definition is motivated by a proof of Sperner's theorem that uses symmetric chains, and was intended as a possible method of attack on the Middle Two Levels Conjecture. We show that the subset lattices have the HC-SCP property by showing that the class of posets with the strong HC-SCP property, a slight strengthening of the HC-SCP property, is closed under cartesian product with a two-element chain. Furthermore, we show that the cartesian product of any two posets from this strong class has the (weak) HC-SCP property.

Submitted by Liliane Ferreira (ljuvencia30@gmail.com) on 2018-05-21T15:48:27Z No. of bitstreams: 2 Dissertação - Vinícius de Sousa Coelho - 2018.pdf: 1071318 bytes, checksum: fba98fd6feb916f0400af915d4d92a2b (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-05-22T12:14:46Z (GMT) No. of bitstreams: 2 Dissertação - Vinícius de Sousa Coelho - 2018.pdf: 1071318 bytes, checksum: fba98fd6feb916f0400af915d4d92a2b (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-05-22T12:14:46Z (GMT). No. of bitstreams: 2 Dissertação - Vinícius de Sousa Coelho - 2018.pdf: 1071318 bytes, checksum: fba98fd6feb916f0400af915d4d92a2b (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-04-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The Maximum Weight Planar Subgraph Problem (MWPSP) consists of identifying a planar subgraph of maximum weight of a given edge-weighted graph. This work proposes new heuristic solutions, mainly using Graphic Processing Units, based on local transformations on the graph topology, consisting of vertex and edge insertion/relocation moves. Sequential and parallel implementations were built and applied to various numerical instances with promising results. One of the approaches requires only 25 seconds of execution, being more than 200 times faster than its corresponding sequential version, for a 100-vertex instance. In terms of quality, the proposed solutions obtained better results than state of the art proposals. / O problema do subgrafo planar de peso máximo (MWPSP) consiste em extrair um subgrafo planar maximal, a partir de um grafo completo com pesos atribuídos às arestas, cuja soma dos pesos das arestas seja máxima. Este trabalho propõe soluções heurísticas, construídas por meio de Unidades de Processamento Gráfico (GPUs), baseadas em transformações locais na topologia do grafo através da inserção/realocação de vértices e arestas. Implementações sequencias e paralelas foram propostas, apresentando resultados satisfatórios. Em uma das propostas, a versão paralela requer cerca de 25 segundos de execução em uma instância de 100 vértices, sendo cerca de 200 vezes mais rápida que a versão sequencial correspondente. Em termos de qualidade da solução, as propostas superaram os resultados obtidos por algoritmos no estado da arte.