Graph Drawing

This paper describes two algorithms for determining the satisfiability of Boolean conjunctive normal form expressions limited to two literals per clause (2-SAT) extending the classic effort of Aspvall, Plass, and Tar- jan. The first algorithm differs from the original in that satisfiability is determined upon the presentation of each clause rather than the entire clause set. This on- line algorithm experimentally exhibits average run-time linear to the number of variables. This performance is achieved by performing a single depth first search of one of the incoming literals. Additional search is avoided by excluding clauses containing pure variables or variables whose truth value has been explicitly pro- vided or can be inferred. An off-line algorithm is also described that incorporates these strategies.

Network visualization is one of the most widely used tools in digital humanities research. The idea of uncertain or "fuzzy" data is also a core notion in digital humanities research. Yet network visualizations in digital humanities do not always prominently represent uncertainty. In this article, we present a mathematical and logical model of uncertainty as a range of values which can be used in network visualizations. We review some of the principles for visualizing uncertainty of di erent kinds, visual variables that can be used for representing uncertainty, and how these variables have been used to represent di erent data types in visualizations drawn from a range of non-humanities fields like climate science and bioinformatics. We then provide examples of two diagrams: one in which the variables displaying degrees of uncertainty are integrated/pinto the graph and one in which glyphs are added to represent data certainty and uncertainty. Finally, we discuss how probabilistic data and what-if scenarios could be used to expand the representation of uncertainty in humanities network visualizations.

Comprehending instructional texts often requires the integration of verbal and visual information. Visual information (especially diagrams) can be helpful for relating and integrating pieces of information and can therefore contribute to mental model building. However, they often fail in achieving any contribution to the instructional process, because of a number of factors. In fact, readers often neglect illustrations and rely too much on textual information. Even when they pay attention to pictorial information, learners often obtain the overall meaning of pictures the easiest and fastest way possible, through low-level processing that gives the learner the illusion of having understood. Moreover, graphs used in textbooks may be difficult to understand, because they often assume an information-telling perspective that does not help readers understand a text's most important data and numerical trend. A further problem is that texts and graphs are often placed far apart from each other and therefore require the learner to scan the page in search of a diagram corresponding to a printed sentence. The aim of this investigation is to verify whether the use of two well-known study strategies (writing-tolearn and drawing visual organizers, i.e., graph drawing) can stimulate the integration of a text's propositions with visual information. Forty-two University students were asked to read a newspaper article (containing one written text and six diagrams) and either to prepare a written reformulation of the text or to draw a visual organizer summarizing the textual information. An analysis of the participants' written notes (i.e., an external and controllable product of task instructions) and of their results in a delayed test provided us with information on the usefulness of the two strategies for information integration and for the monitoring of text comprehension. An analysis of overall group main effects revealed significant differences among instructional conditions for total scores and for scores on diagram and integration questions, but not for text questions. The group of students who drew graphs outperformed the others (the writing-to-learn and reading-only groups) on all measures requiring the integration of textual and graphic information.

The edge-reconstruction number ern(G) of a graph G is equal to the minimum number of edge-deleted subgraphs G-e of G which are sufficient to determine G up to isomorphism. Building upon the work of Molina and using results from computer searches by Rivshin and more recent ones which we carried out, we show that, apart from three known exceptions, all bicentroidal trees have edge-reconstruction number equal to 2. We also exhibit the known trees having edge-reconstruction number equal to 3 and we conjecture that the three infinite families of unicentroidal trees which we have found to have edge-reconstruction number equal to 3 are the only ones.

In this paper, we present a case study for the visualisation and analysis of large and complex temporal multivariate networks derived from the Internet Movie DataBase (IMDB). Our approach is to integrate network analysis methods with visualisation in order to address scalability and complexity issues. In particular, we defined new analysis methods such as (p,q)-core and 4-ring to identify important dense subgraphs and short cycles from the huge bipartite graphs. We applied island analysis for a specific time slice in order to identify important and meaningful subgraphs. Further, a temporal Kevin Bacon graph and a temporal two mode network are extracted in order to provide insight and knowledge on the evolution.

Graph visualizations increase the perception of entity relationships in a network. However, as graph size and density increases, readability rapidly diminishes. In this article, we present an end-to-end, tile-based visual analytic approach called graph mapping that utilizes cluster computing to turn large-scale graph (node–link) data into interactive visualizations in modern web browsers. Our approach is designed for end-user analysis of community structure and relationships at macro- and micro scales. We also present the results of several experiments using alternate methods for qualitatively improving comprehensibility of hierarchical community detection visualizations by proposing constraints to state-of-the-art modularity maximization algorithms.

In this paper, we present in-progress work on "Influent", a graph analysis tool that enables an intelligence analyst to visually and interactively "follow the money" or other transaction flow. Summary visualizations of transactional patterns and entity characteristics, a left-to-right semantic flow layout, interactive link expansion and hierarchical entity clustering enable Influent to operate effectively at scale with millions of entities and hundreds of millions of transactions, with larger data sets in progress.

Gene regulatory networks inferred from RNA abundance data have generated significant interest, but despite this, gene network approaches are used infrequently and often require input from bioinformaticians. We have assembled a suite of tools for analysing regulatory networks, and we illustrate their use with microarray datasets generated in human endothelial cells. We infer a range of regulatory networks, and based on this analysis discuss the strengths and limitations of network inference from RNA abundance data. We welcome contact from researchers interested in using our inference and visualization tools to answer biological questions.

We describe here a collection of heuristics for producing "nice"-drawings of directed graphs, and a simple dual-mode software tool for testing and evaluating them. In playing mode, the heuristics are applied in random sequence over a set of drawings, in the manner of an asynchronous team (A-team). As new drawings are added to the set, others are deleted based on a multi-valued aesthetic evaluation function. By inspecting the "pedigree" of the best solutions found in playing mode, the user can obtain insights into the best order in which the heuristics should be applied. Then the user can test these insights in the working mode, where the heuristics are applied in a fixed sequence. Some of the heuristics that we describe here are similar to the steps of Sugiyama's D-ABDUCTOR graph-drawing package; and indeed we can obtain results similar to those of D-ABDUCTOR, by applying our heuristics in the proper sequence.

In this paper,  we investigate the structural properties of trees and bipartite graphs through the lens of topological indices and combinatorial graph theory. We focus on the First and Second Hyper-Zagreb indices, $HM_1(G)$ and $HM_2(G)$, for trees $T \in T(n, \Delta)$ with $n$ vertices and maximum degree $\Delta$. Key propositions demonstrate that the presence of end-support or support vertices of degree at least three, distinct from a vertex of maximum degree, implies the existence of another tree $T' \in T(n, \Delta)$ with strictly smaller Hyper-Zagreb indices. These results are extended to exponential forms, highlighting the influence of high-degree vertices. Additionally, we explore structural characterizations of trees via degree sequence majorization and $S$-order, establishing conditions for the first and last trees in specific classes. For bipartite graphs, we examine equitable coloring, cycle lengths, and $k$-redundant tree embeddings, supported by theorems on connectivity and minimum degree constraints. The paper also addresses the independence number of bipartite graphs, exterior covers, and competition numbers of complete $r$-partite graphs, providing bounds and structural insights. Finally, we discuss Markov-chain algorithms for generating bipartite graphs and tournaments with prescribed degree sequences, analyzing their mixing times and convergence properties. These results contribute to the understanding of extremal properties and combinatorial structures in graph theory, with applications in chemical graph theory and network analysis.

The grid obstacle representation, or alternately, 1 -obstacle representation of a graph G = (V, E) is an injective function f : V → Z 2 and a set of point obstacles O on the grid points of Z 2 (where no vertex of V has been mapped) such that uv is an edge in G if and only if there exists a Manhattan path between f (u) and f (v) in Z 2 avoiding the obstacles of O and points in f (V). This work shows that planar graphs admit such a representation while there exist some non-planar graphs that do not admit such a representation. Moreover, we show that every graph admits a grid obstacle representation in Z 3 . We also show NP-hardness result for the point set embeddability of an 1 -obstacle representation.

We present new O(n)-time methods for planar embedding and Kuratowski subgraph isolation that were inspired by the Booth-Lueker PQ-tree implementation of the Lempel-Even-Cederbaum vertex addition method. In this paper, we improve upon our conference proceedings formulation and upon the Shih-Hsu PC-tree, both of which perform comprehensive tests of planarity conditions embedding the edges from a vertex to its descendants in a 'batch' vertex addition operation. These tests are simpler than but analogous to the templating scheme of the PQ-tree. Instead, we take the edge to be the fundamental unit of addition to the partial embedding while preserving planarity. This eliminates the batch planarity condition testing in favor of a few localized decisions of a path traversal process, and it exploits the fact that subgraphs can become biconnected by adding a single edge. Our method is presented using only graph constructs, but our definition of external activity, path traversal process and theoretical analysis of correctness can be applied to optimize the PC-tree as well.

A graph is planar if it can be drawn on the plane with no crossing edges. There are several linear time planar embedding algorithms but ah are considered by many to be quite complicated. This paper presents a new method for performing linear time planar graph embedding which avoids some of the complexities of previous approaches (including the need to first st-number the vertices). Our new algorithm easily permits the extraction of a planar obstruction (a subgraph homeomorphic to Ks,s or Ks) in O(n) time if the graph is not planar. Lueker which uses a data structure called a PQ-tree. The Pnodes in a P&-tree represent parts of the partially embedded graph that can be permuted, and the Q-nodes represent parts that can be flipped. We avoid the use of P-nodes by not connecting pieces together until they become biconnected. We avoid Q nodes by using a data structure which allows biconnected components to be fiipped in O(1) time.

Graph sampling methods have been used to reduce the size and complexity of big complex networks for graph mining and visualization. However, existing graph sampling methods often fail to preserve the connectivity and important structures of the original graph. This paper introduces a new divide and conquer approach to spectral graph sampling based on graph connectivity, called the BC Tree (i.e., decomposition of a connected graph into biconnected components) and spectral sparsification. Specifically, we present two methods, spectral vertex sampling$$BC\_SV$$BC_SVand spectral edge sampling$$BC\_SS$$BC_SSby computing effective resistance values of vertices and edges for each connected component. Furthermore, we present$$DBC\_SS$$DBC_SSand$$DBC\_GD$$DBC_GD, graph connectivity-based distributed algorithms for spectral sparsification and graph drawing respectively, aiming to further improve the runtime efficiency of spectral sparsification and graph drawing by integrating connectivity-ba...

Network structures (graphs) have become a natural part of everyday life and their analysis helps to gain an understanding of their inherent structure and the real-world aspects thereby expressed. The exploration of graphs is largely supported and driven by visual means. The aim of this thesis is to give a comprehensive view on the problems associated with these visual means and to detail concrete solution approaches for them. This is done on all three levels involved: the data level, the representation level, and the task level. In a first step, the intricate dependencies on the representation level alone are discussed for the case of implicit tree visualizations. With the awareness of the representation issues involved, a second step takes the characteristics on data level into account, which are most importantly graph size and graph type. Their influence on the visualization design are discussed specifically for large trees and bipartite graphs. Finally, with bringing in the task level, the entire exploration workflow from computational preprocessing to the interaction with the visualization can be captured and thus discussed in terms of its consequences for the visualization design. This includes issues of necessary confirmative elements in the exploratory analysis, as well as selecting appropriate data sets from a heterogeneous data pool for an analysis. Concrete visualization techniques are introduced to underline the value of this comprehensive discussion for supporting explorative graph visualization.

Set visualization is a well-known task in information visualization. In biology, it is used for comparing visually sets of genes or proteins, typically using Venn diagrams. However, limitations of the Venn diagram are well-known: they are limited to 6 sets and difficult to read above 4. Many other set visualization techniques have been proposed, but they have never been widely used in biology. In this paper, we introduce RainBio, a technique for visualizing sets in biology and aimed at providing a global overview showing the size of the main intersections, in a proportional way, and the similarities between sets. We adapt rainbow boxes, a technique for visualizing small datasets, to the visualization of larger sets, using element aggregation and intersection clustering. We present the application of RainBio to three datasets, with 5, 6 and 12 sets. We also describe a small user study comparing RainBio with Venn diagrams, involving 30 students in biology. Results showed that RainBio led to significantly fewer errors on 6-set dataset, and that the majority of students preferred RainBio. RainBio is proposed as a web-based tool for up to 15 sets.

The reload cost concept refers to the cost that occurs at a vertex along a path on an edge-colored graph when it traverses an internal vertex between two edges of different colors. This cost depends only on the colors of the traversed edges. Reload costs arise in various applications such as transportation networks, energy distribution networks, and telecommunications. Previous work on reload costs focuses on two problems of finding a spanning tree with minimum cost with respect to two different cost measures. In both problems the cost is associated with a set of paths from a given vertex r to all the leaves of the constructed tree. The first cost measure is the sum of the reload costs of all paths from r to the leaves. The second cost measure is the changeover cost, in which the cost of traversing a vertex by using two specific incident edges is paid only once regardless of the number of paths traversing it. The first problem is inapproximable within any polynomial time computable function of the input size [1], and the second problem is inapproximable within n 1-for any > 0 . In this paper we show that the first hardness result holds also for the second problem. Given this strong inapproximability result, we study the complexity and approximability properties of numerous special cases of this second problem. We mainly focus on bounded costs, and consider both directed and undirected graphs, bounded and unbounded number of colors, and both bounded and unbounded degree graphs. We also present polynomial time exact algorithms and an approximation algorithm for some special case. To the best of our knowledge, these are the first algorithms with a provable performance guarantee for the problem. Moreover, our approximation algorithm shows a tight bound on the approximability of the problem for a specific family of instances.

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide-to the best of our knowledge-the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.

We study efficient combinatorial algorithms to produce the Hasse diagram of the poset of bounded faces of an unbounded polyhedron, given vertex-facet incidences. We also discuss the special case of simple polyhedra and present computational results.

This paper introduces compressed certificates for planarity, biconnectivity and triconnectivity in planar graphs, and proves many structural properties of certificates in planar graphs. As an application of our compressed certificates, we develop efficient dynamic planar algorithms. In particular, we consider the following three operations on a planar graph G : (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without violating planarity. We show how to support each of the above operations in O ( n 2/3 ) time, where n is the number of vertices in the graph. The bound for tests and deletions is worst-case, while the bound for insertions is amortized. This is the first algorithm for this problem with sub-linear running time, and it affirmatively answers a question posed in Epstein et al. [1992]. We use our compressed certificates for biconnectivity and triconnectivity to maintain the biconnected an...

The study of evolution of networks has received increased interest with the recent discovery that many real-world networks possess many things in common, in particular the manner of evolution of such networks. By adding a dimension of time to graph analysis, evolving graphs present opportunities and challenges to extract valuable information. This paper introduces the Evolving Graph Markup Language (EGML), an XML application for representing evolving graphs and related results. Along with EGML, a software tool is provided for the study of evolving graphs. New evolving graph drawing techniques based on the force-directed graph layout algorithm are also explored. Our evolving graph techniques reduce vertex movements between graph instances, so that an evolving graph can be viewed with smooth transitions.

In this paper, we study the complexity of rectangular cartograms, i.e., maps where every region is a rectangle, and which should be deformed such that given area requirements are satisfied. We study the closely related problem of cartograms with orthogonal octagons, and show that this problem is NP-hard. From our proof, it also follows that rectangular cartograms are NP-hard if we allow for the existence of a "sea", i.e., a region of arbitrarily high complexity on the outside of the drawing.

Excessive edge density in graphs can cause serious readability issues, which in turn can make the graphs difficult to understand or even misleading. Recently, we introduced the idea of providing tools that offer interactive edge bending as a method by which edge congestion can be disambiguated. We extend this direction, presenting a new tool, Edge Plucking, which offers new interactive methods to clarify node-edge relationships. Edge Plucking expands the number of situations in which interactive graph exploration tools can be used to address edge congestion.

When exploiting the power of node-link diagrams to represent real-world data such as web structures, airline routes, electrical, telecommunication and social networks, link congestion frequently arises. Such areas in the diagram-with dense, overlapping links-are not readable: connectivity, node shapes, labels, and contextual information are obscured. In response, graph-layout research has begun to consider the modification of link shapes with techniques such as link routing and bundling. In this paper, we delve into the interactive techniques afforded by variant use of link curvature, delineating a six-dimensional design space that is populated by four families of interactive techniques: bundling, fanning, magnets, and legends. Our taxonomy encompasses existing techniques and reveals several novel link interactions. We describe the implementation of these techniques and illustrate their potential for exploring dense graphs with multiple types of links.

We address the unsolved problem of unfolding prismatoids in a new context, viewing a "topless prismatoid" as a convex patch-a polyhedral subset of the surface of a convex polyhedron homeomorphic to a disk. We show that several natural strategies for unfolding a prismatoid can fail, but obtain a positive result for "petal unfolding" topless prismatoids. We also show that the natural extension to a convex patch consisting of a face of a polyhedron and all its incident faces, does not always have a nonoverlapping petal unfolding. However, we obtain a positive result by excluding the problematical patches. This then leads a positive result for restricted prismatoids. Finally, we suggest suggest studying the unfolding of convex patches in general, and offer some possible lines of investigation.

Background: The advent of "omics" science has brought new perspectives in contemporary biology through the high-throughput analyses of molecular interactions, providing new clues in protein/gene function and in the organization of biological pathways. Biomolecular interaction networks, or graphs, are simple abstract representations where the components of a cell (e.g. proteins, metabolites etc.) are represented by nodes and their interactions are represented by edges. An appropriate visualization of data is crucial for understanding such networks, since pathways are related to functions that occur in specific regions of the cell. The force-directed layout is an important and widely used technique to draw networks according to their topologies. Placing the networks into cellular compartments helps to quickly identify where network elements are located and, more specifically, concentrated. Currently, only a few tools provide the capability of visually organizing networks by cellular compartments. Most of them cannot handle large and dense networks. Even for small networks with hundreds of nodes the available tools are not able to reposition the network while the user is interacting, limiting the visual exploration capability. Results: Here we propose CellNetVis, a web tool to easily display biological networks in a cell diagram employing a constrained force-directed layout algorithm. The tool is freely available and open-source. It was originally designed for networks generated by the Integrated Interactome System and can be used with networks from others databases, like InnateDB. Conclusions: CellNetVis has demonstrated to be applicable for dynamic investigation of complex networks over a consistent representation of a cell on the Web, with capabilities not matched elsewhere.

Figure 1: Mobile devices support graph visualization and interaction on wall-sized displays close to the display wall and further away (A). The GRASP system provides a mobile toolbox with selections, alternative representations, lenses, and filtering close to the user (B).

A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn as line segments between the layers. In this paper we study the 2-Layer Planarization problem: can k edges be deleted from a given graph G so that the remaining graph is biplanar? This problem is N P-complete, and remains so if the permutation of the vertices in one layer is fixed (the 1-Layer Planarization problem). We prove that these problems are fixed-parameter tractable by giving linear-time algorithms for their solution (for fixed k). In particular, we solve the 2-Layer Planarization problem in O(k • 6 k + |G|) time and the 1-Layer Planarization problem in O(3 k • |G|) time. We also show that there are polynomial-time constant-approximation algorithms for both problems.

This paper presents a new learning tool developed for the visualization of general directed and undirected rooted trees. This visualization tool for teaching graph and network algorithms provides an interactive view of the subject being taught to the students. The tool is designed for three different groups of users: developers, instructors and students. It could be used for the visualization of any algorithm that generates a sequence of trees. This new tool minimizes the instructor’s effort and gives him the possibility of drawing trees easily with many different properties. It can be used efficiently in courses like Graph Theory or Network Programming or in an introductory course like Algorithms and Data Structures. We believe that this tool will be an effective supplement to traditional classroom education. It is a commonly accepted fact that teaching can be vastly amplified when it is done visually and interactively instead of in the traditional way. Therefore, a summary of othe...

We investigate crossing minimization problems for a set of permutations, where a crossing expresses a disarrangement between elements. The goal is a common permutation π * which minimizes the number of crossings. This is known as the Kemeny optimal aggregation problem minimizing the Kendall-τ distance. Recent interest into this problem comes from application to meta-search and spam reduction on the Web. This rank aggregation problem can be phrased as a one-sided twolayer crossing minimization problem for an edge coloured bipartite graph, where crossings are counted only for monochromatic edges. Here we introduce the max version of the crossing minimization problem, which attempts to minimize the discrimination against any permutation. We show the NP-hardness of the common and the max version for k ≥ 4 permutations (and k even), and establish a 2-2/k and a 2-approximation, respectively. For two permutations crossing minimization is solved by inspecting the drawings, whereas it remains open for three permutations.

Keçeci Layout is a deterministic node layout algorithm designed for graph visualization in Python. Its primary purpose is to position the nodes of a graph in a predefined, sequential, and repeatable manner. The algorithm processes nodes sequentially, placing them along a user-defined primary axis (e.g., top-down or left-to-right) while applying an offset on the secondary axis in a zigzag pattern. This zigzag pattern helps prevent node overlaps while maintaining an orderly structure. The function is designed to be compatible with popular Python graph libraries, including NetworkX, Rustworkx, igraph, Networkit, and Graphillion (via GraphSet objects). It takes a graph object from one of these libraries as input, processes the nodes (usually by sorting their IDs), and returns a Python dictionary mapping each node's identifier (in the library-specific format) to its calculated (x, y) coordinates. Users can customize the node spacing along the primary and secondary axes (`primary_spacing`, `secondary_spacing`), the main direction of the layout (`primary_direction`), and the starting side of the zigzag pattern (`secondary_start`) through parameters. This regular and predictable structure is useful, particularly when the order of nodes is significant or when a simple, aesthetically pleasing, and easily traceable graph visualization is desired. Keçeci Layout is a deterministic node layout algorithm designed for graph visualization in Python. Its primary purpose is to position the nodes of a graph in a predefined, sequential, and repeatable manner. The algorithm processes nodes sequentially, placing them along a user-defined primary axis (e.g., top-down or left-to-right) while applying an offset on the secondary axis in a zigzag pattern. This zigzag pattern helps prevent node overlaps while maintaining an orderly structure. The function is designed to be compatible with popular Python graph libraries, including NetworkX, Rustworkx, igraph, Networkit, and Graphillion (via GraphSet objects). It takes a graph object from one of these libraries as input, processes the nodes (usually by sorting their IDs), and returns a Python dictionary mapping each node's identifier (in the library-specific format) to its calculated (x, y) coordinates. Users can customize the node spacing along the primary and secondary axes (`primary_spacing`, `secondary_spacing`), the main direction of the layout (`primary_direction`), and the starting side of the zigzag pattern (`secondary_start`) through parameters. This regular and predictable structure is useful, particularly when the order of nodes is significant or when a simple, aesthetically pleasing, and easily traceable graph visualization is desired.

We present CiteWiz, an extensible framework for visualization of scientific citation networks. The system is based on a taxonomy of citation database usage for researchers, and provides a timeline visualization for overviews and an influence visualization for detailed views. The timeline displays the general chronology and importance of authors and articles in a citation database, whereas the influence visualization is implemented using the Growing Polygons technique, suitably modified to the context of browsing citation data. Using the latter technique, hierarchies of articles with potentially very long citation chains can be graphically represented. The visualization is augmented with mechanisms for parent–child visualization and suitable interaction techniques for interacting with the view hierarchy and the individual articles in the dataset. We also provide an interactive concept map for keywords and co-authorship using a basic force-directed graph layout scheme. A formal user s...

Graphs arise in a natural way in many applications, together with the need to be drawn. Except for very small instances, drawing a graph by hand becomes a very complex task, which must be performed by automatic tools. The field of graph drawing is concerned with finding algorithms to draw graph in an aesthetically pleasant way, based upon a certain number of aesthetic criteria that define what a good drawing, (synonyms: diagrams, pictures, layouts), of a graph should be. This problem can be found in many such as in the computer networks, data networks, class inter-relationship diagrams in object oriented databases and object oriented programs, visual programming interfaces, database design systems, software engineering…etc. Given a plane graph G, we wish to find a drawing of G in the plane such that the vertices of G are represented as grid points, and the edges are represented as straight-line segments between their endpoints without any edgeintersection. Such drawings are called planar straight-line drawings of G. An additional objective is to minimize the area of the rectangular grid in which G is drawn. In this paper we introduce a new algorithms that finds an embedding of 3-planar graph.

The field of graph drawing is concerned with finding algorithms to draw graph in an aesthetically pleasant way, based upon a certain number of aesthetic criteria that define what a good drawing, (synonyms: diagrams, pictures, layouts), of a graph should be. This problem can be found in many such as in the computer networks, data networks, class interrelationship diagrams in object oriented databases and object oriented programs, visual programming interfaces, database design systems, software engineering…etc. Given a plane graph G, we wish to find a drawing of G in the plane such that the vertices of G are represented as grid points, and the edges are represented as straight-line segments between their endpoints without any edge-intersection. Such drawings are called planar straight-line drawings of G. An additional objective is to minimize the area of the rectangular grid in which G is drawn. In this paper we introduce a new algorithms that finds an embedding of 3-planar graph in linear time O(n).

The field of graph drawing is concerned with finding algorithms to draw graph in an aesthetically pleasant way, based upon a certain number of aesthetic criteria that define what a good drawing, (synonyms: diagrams, pictures, layouts), of a graph should be. This problem can be found in many such as in the computer networks, data networks, class interrelationship diagrams in object oriented databases and object oriented programs, visual programming interfaces, database design systems, software engineering…etc. Given a plane graph G, we wish to find a drawing of G in the plane such that the vertices of G are represented as grid points, and the edges are represented as straight-line segments between their endpoints without any edge-intersection. Such drawings are called planar straight-line drawings of G. An additional objective is to minimize the area of the rectangular grid in which G is drawn. In this paper we introduce a new algorithms that finds an embedding of 3-planar graph in linear time O(n).

Graphs arise in a natural way in many applications, together with the need to be drawn. Except for very small instances, drawing a graph by hand becomes a very complex task, which must be performed by automatic tools. The field of graph drawing is concerned with finding algorithms to draw graph in an aesthetically pleasant way, based upon a certain number of aesthetic criteria that define what a good drawing, (synonyms: diagrams, pictures, layouts), of a graph should be. This problem can be found in many such as in the computer networks, data networks, class inter-relationship diagrams in object oriented databases and object oriented programs, visual programming interfaces, database design systems, software engineering…etc. Given a plane graph G, we wish to find a drawing of G in the plane such that the vertices of G are represented as grid points, and the edges are represented as straight-line segments between their endpoints without any edgeintersection. Such drawings are called planar straight-line drawings of G. An additional objective is to minimize the area of the rectangular grid in which G is drawn. In this paper we introduce a new algorithms that finds an embedding of 3-planar graph.

Graphs are very effective tools in visualizing information and are used in many fields including the medical field. In most developing countries primary care, graphs are used to monitor child growth. These measures are therefore often displayed using line graphs, basing it on three indicators (stunting, underweight and wasting) based on the WHO 2006 Child Growth Standard. Most literature on information visualization of electronic health record data focuses on aggregate data visualization tools. This research therefore, was set out to provide such an overview of requirements for computerized graphs for individual patient data, implemented in a way that all kinds of medical graphs showing the development of medical measures over time can be displayed. This research was interpretive, using a user-centric approach for data collection where interviews and web search was used to ensure that the graphs developed are fit the user requirements. This followed prototype development using one o...

The Reactome Knowledgebase (www.reactome.org) provides molecular details of signal transduction, transport, DNA replication, metabolism and other cellular processes as an ordered network of molecular transformations--an extended version of a classic metabolic map, in a single consistent data model. Reactome functions both as an archive of biological processes and as a tool for discovering unexpected functional relationships in data such as gene expression pattern surveys or somatic mutation catalogues from tumour cells. Over the last two years we redeveloped major components of the Reactome web interface to improve usability, responsiveness and data visualization. A new pathway diagram viewer provides a faster, clearer interface and smooth zooming from the entire reaction network to the details of individual reactions. Tool performance for analysis of user datasets has been substantially improved, now generating detailed results for genome-wide expression datasets within seconds. The analysis module can now be accessed through a RESTFul interface, facilitating its inclusion in third party applications. A new overview module allows the visualization of analysis results on a genome-wide Reactome pathway hierarchy using a single screen page. The search interface now provides auto-completion as well as a faceted search to narrow result lists efficiently.

While graph drawing focuses more on the aesthetic representation of node-link diagrams, graph visualization takes into account other visual metaphors making them useful for graph exploration tasks in information visualization and visual analytics. Although there are aesthetic graph drawing criteria that describe how a graph should be presented to make it faster and more reliably explorable, many controlled and uncontrolled empirical user studies flourished over the past years. The goal of them is to uncover how well the human user performs graph-specific tasks, in many cases compared to previously designed graph visualizations. Due to the fact that many parameters in a graph dataset as well as the visual representation of them might be varied and many user studies have been conducted in this space, a state-of-the-art survey is needed to understand evaluation results and findings to inform the future design, research, and application of graph visualizations. In this article, we classify the present literature on the topmost level into graph interpretation, graph memorability, and graph creation where the users with their tasks stand in focus of the evaluation, not the computational aspects. As another outcome of this work, we identify the white spots in this field and sketch ideas for future research directions.

Given a simple polygon $ \mathcal {P} $ of $ n $ vertices in the Plane. We study the problem of computing the visibility area from a given viewpoint $ q $ inside $ \mathcal {P} $ where only sub-linear variables are allowed for working space. Without any memory-constrained, this problem was previously solved in $ O(n) $-time and $ O(n) $-variables space. In a newer research, the visibility area of a point be computed in $ O(n) $-time, using $ O(\sqrt{n}) $ variables for working space. In this paper, we present an optimal-time algorithm, using $ O(c/\log n) $ variables space for computing visibility area, where $ c<n $ is the number of critical vertices. We keep the algorithm in the linear-time and reduce space as much as possible.

An orthogonal drawing of a planar graph G is a drawing of G such that each vertex is drawn as a point, each edge is drawn as a sequence of alternate horizontal and vertical line segment, and any two edges do not cross except at their common end. An L-shaped drawing of a graph G is an orthogonal drawing of G where each edge has exactly one bend. Clearly the maximum degree of a graph G is at most 4 if G has an L-shaped drawing. It is known that every planar graph of maximum degree 3 does not have L-shaped drawing. In this paper we study Lshaped drawings of series-parallel graphs in variable embedding settings. We show that every series-parallel graph of maximum degree 3 and every maximal outerplane graph of maximum degree 4 admit L-shaped drawings.

The paper presents self-organizing graphs, a novel approach to graph layout based on a competitive learning algorithm. This method is an extension of self-organization strategies known from unsupervised neural networks, namely from Kohonen's self-organizing map. Its main advantage is that it is very flexibly adaptable to arbitrary types of visualization spaces, for it is explicitly parameterized by a metric model of the layout space. Yet the method consumes comparatively little computational resources and does not need any heavy-duty preprocessing. Unlike with other stochastic layout algorithms, not even the costly repeated evaluation of an objective function is required. To our knowledge this is the first connectionist approach to graph layout. The paper presents applications to 2D-layout as well as to 3D-layout and to layout in arbitrary metric spaces, such as networks on spherical surfaces.