Design gallery browsers based on 2D and 3D graph drawing

Many problems in computer-aided design and graphics involve the process of setting and adjusting input parameters to obtain desirable output values. Exploring different parameter settings can be a difficult and tedious task in most such systems. In the Design GalleryTM (DG) approach, parameter setting is made easier by dividing the task more equitably between user and computer. DG interfaces present the user with the broadest selection, automatically generated and organized, of perceptually different designs that can be produced by varying a given set of input parameters. The DG approach has been applied to several difficult parameter-setting tasks from the field of computer graphics: light selection and placement for image rendering; opacity and color transfer-function specification for volume rendering; and motion control for articulated-figure and particle-system animation. The principal technical challenges posed by the DG approach are dispersion (finding a set of input-parameter vectors that optimally disperses the resulting output values) and arrangement (arranging the resulting designs for easy browsing by the user). We show how effective arrangement can be achieved with 2D and 3D graph drawing. While navigation is easier in the 2D interface, the 3D interface has proven to be surprisingly usable, and the 3D drawings sometimes provide insights that are not so obvious in the 2D drawings.Engineering and Applied Science

Stack and Queue Layouts via Layered Separators

It is known that every proper minor-closed class of graphs has bounded stack-number (a.k.a. book thickness and page number). While this includes notable graph families such as planar graphs and graphs of bounded genus, many other graph families are not closed under taking minors. For fixed $g$ and $k$, we show that every $n$-vertex graph that can be embedded on a surface of genus $g$ with at most $k$ crossings per edge has stack-number $\mathcal{O}(\log n)$; this includes $k$-planar graphs. The previously best known bound for the stack-number of these families was $\mathcal{O}(\sqrt{n})$, except in the case of $1$-planar graphs. Analogous results are proved for map graphs that can be embedded on a surface of fixed genus. None of these families is closed under taking minors. The main ingredient in the proof of these results is a construction proving that $n$-vertex graphs that admit constant layered separators have $\mathcal{O}(\log n)$ stack-number.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Drawing a Graph in a Hypercube

A $d$-dimensional hypercube drawing of a graph represents the vertices by distinct points in $\{0,1\}^d$, such that the line-segments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to be related to Sidon sets and antimagic injections.Comment: Submitte

Pixel and Voxel Representations of Graphs

We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for $k$-outerplanar graphs with $n$ vertices, $\Theta(kn)$ pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, $\Theta(n^2)$ voxels are always sufficient and sometimes necessary for any $n$-vertex graph. We improve this bound to $\Theta(n\cdot \tau)$ for graphs of treewidth $\tau$ and to $O((g+1)^2n\log^2n)$ for graphs of genus $g$. In particular, planar graphs admit representations with $O(n\log^2n)$ voxels

Web-based drawing software for graphs in 3D and two layout algorithms

A new web-based software system for visualization and manipulation of graphs in 3D, named We3Graph is presented with a focus on accessibility, customizability for applications of graph drawing, usability and extendibility. The software system allows multiple users to work on the same graph at the same time and is accessible through web browsers. The software can be extended using plugins written in any programming language and custom render engines written in the Javascript language. Also two new algorithms are proposed to answer the following question, previously raised in [53]: Given a graph G with n vertices, V = fv1;v2; : : : ;vng, and given a set of n distinct points P = fp1; p2; : : : ; png each with integer coordinates in three dimensions, can G be drawn crossing-free on P with vi at pi and with a number of bends polynomial in n and in a volume polynomial in n and the dimension of P

Layout of Graphs with Bounded Tree-Width

A \emph{queue layout} of a graph consists of a total order of the vertices, and a partition of the edges into \emph{queues}, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its \emph{queue-number}. A \emph{three-dimensional (straight-line grid) drawing} of a graph represents the vertices by points in $\mathbb{Z}^3$ and the edges by non-crossing line-segments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of three-dimensional drawing of a graph $G$ is closely related to the queue-number of $G$. In particular, if $G$ is an $n$-vertex member of a proper minor-closed family of graphs (such as a planar graph), then $G$ has a $O(1)\times O(1)\times O(n)$ drawing if and only if $G$ has O(1) queue-number. (2) It is proved that queue-number is bounded by tree-width, thus resolving an open problem due to Ganley and Heath (2001), and disproving a conjecture of Pemmaraju (1992). This result provides renewed hope for the positive resolution of a number of open problems in the theory of queue layouts. (3) It is proved that graphs of bounded tree-width have three-dimensional drawings with O(n) volume. This is the most general family of graphs known to admit three-dimensional drawings with O(n) volume. The proofs depend upon our results regarding \emph{track layouts} and \emph{tree-partitions} of graphs, which may be of independent interest.Comment: This is a revised version of a journal paper submitted in October 2002. This paper incorporates the following conference papers: (1) Dujmovic', Morin & Wood. Path-width and three-dimensional straight-line grid drawings of graphs (GD'02), LNCS 2528:42-53, Springer, 2002. (2) Wood. Queue layouts, tree-width, and three-dimensional graph drawing (FSTTCS'02), LNCS 2556:348--359, Springer, 2002. (3) Dujmovic' & Wood. Tree-partitions of $k$-trees with applications in graph layout (WG '03), LNCS 2880:205-217, 200

Un guide sur la toile pour sélectionner un logiciel de tracé de graphes

Nouvelle adresse du site : http://gvsr.polytech.univ-nantes.frNational audienceLes graphes permettent d'une part, en tant qu'objets combinatoires, de modéliser et d'analyser des systèmes de relations complexes entre des entités et d'autre part, de représenter ces relations sur des supports visuels afin de les rendre accessibles à un utilisateur non spécialiste. Si ces derniers sont rapidement devenus des outils privilégiés pour de nombreuses problématiques applicatives comme la découverte de relations non explicites en veille technologique, le choix d'une méthode de visualisation efficace reste encore très souvent une question ouverte pour l'utilisateur. Afin de guider ce dernier dans sa sélection, nous dressons une typologie succincte des principaux modes de représentation à savoir le tracé statique, le tracé interactif et le tracé des très grands graphes, et nous finissons par la présentation d'un nouveau site Web, dédié au référencement des logiciels de tracés de graphes. Le site est consultable à l'adresse : http://hulk.knowesis.fr/GVSR http://gvsr.polytech.univ-nantes.fr. Son originalité réside en particulier dans une présentation homogène des informations pertinentes mises en œuvre au travers d'un ensemble de fiches codées en XML

Visualisation of BioPAX Networks using BioLayout Express (3D).

BioLayout Express (3D) is a network analysis tool designed for the visualisation and analysis of graphs derived from biological data. It has proved to be powerful in the analysis of gene expression data, biological pathways and in a range of other applications. In version 3.2 of the tool we have introduced the ability to import, merge and display pathways and protein interaction networks available in the BioPAX Level 3 standard exchange format. A graphical interface allows users to search for pathways or interaction data stored in the Pathway Commons database. Queries using either gene/protein or pathway names are made via the cPath2 client and users can also define the source and/or species of information that they wish to examine. Data matching a query are listed and individual records may be viewed in isolation or merged using an 'Advanced' query tab. A visualisation scheme has been defined by mapping BioPAX entity types to a range of glyphs. Graphs of these data can be viewed and explored within BioLayout as 2D or 3D graph layouts, where they can be edited and/or exported for visualisation and editing within other tools