Contact Representations of Graphs in 3D

We study contact representations of graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We show that for every 3-connected planar graph, there exists a simultaneous representation of the graph and its dual with 3D boxes. We give a linear-time algorithm for constructing such a representation. This result extends the existing primal-dual contact representations of planar graphs in 2D using circles and triangles. While contact graphs in 2D directly correspond to planar graphs, we next study representations of non-planar graphs in 3D. In particular we consider representations of optimal 1-planar graphs. A graph is 1-planar if there exists a drawing in the plane where each edge is crossed at most once, and an optimal n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a linear-time algorithm for representing optimal 1-planar graphs without separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph admits a representation with boxes. Hence, we consider contact representations with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a quadratic-time algorithm for representing optimal 1-planar graph with L-shaped polyhedra

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

Contact representations of graphs in 3D

We study contact representations of non-planar graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We present a liner-time algorithm constructing a representation of a 3-connected planar graph, its dual, and the vertex-face incidence graph with 3D boxes. We then investigate contact representations of 1- planar graphs. We first prove that optimal 1-planar graphs without separating 4-cycles admit a contact representation with 3D boxes. However, since not every optimal 1-planar graph can be represented in this way, we also consider contact representations with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a quadratic-time algorithm for representing optimal 1-planar graphs with L-shapes. © Springer International Publishing Switzerland 2015

Toward a multilevel representation of protein molecules: comparative approaches to the aggregation/folding propensity problem

This paper builds upon the fundamental work of Niwa et al. [34], which provides the unique possibility to analyze the relative aggregation/folding propensity of the elements of the entire Escherichia coli (E. coli) proteome in a cell-free standardized microenvironment. The hardness of the problem comes from the superposition between the driving forces of intra- and inter-molecule interactions and it is mirrored by the evidences of shift from folding to aggregation phenotypes by single-point mutations [10]. Here we apply several state-of-the-art classification methods coming from the field of structural pattern recognition, with the aim to compare different representations of the same proteins gathered from the Niwa et al. data base; such representations include sequences and labeled (contact) graphs enriched with chemico-physical attributes. By this comparison, we are able to identify also some interesting general properties of proteins. Notably, (i) we suggest a threshold around 250 residues discriminating "easily foldable" from "hardly foldable" molecules consistent with other independent experiments, and (ii) we highlight the relevance of contact graph spectra for folding behavior discrimination and characterization of the E. coli solubility data. The soundness of the experimental results presented in this paper is proved by the statistically relevant relationships discovered among the chemico-physical description of proteins and the developed cost matrix of substitution used in the various discrimination systems.Comment: 17 pages, 3 figures, 46 reference

Constructing sonified haptic line graphs for the blind student: first steps

Line graphs stand as an established information visualisation and analysis technique taught at various levels of difficulty according to standard Mathematics curricula. It has been argued that blind individuals cannot use line graphs as a visualisation and analytic tool because they currently primarily exist in the visual medium. The research described in this paper aims at making line graphs accessible to blind students through auditory and haptic media. We describe (1) our design space for representing line graphs, (2) the technology we use to develop our prototypes and (3) the insights from our preliminary work

TactileGCN: A Graph Convolutional Network for Predicting Grasp Stability with Tactile Sensors

Tactile sensors provide useful contact data during the interaction with an object which can be used to accurately learn to determine the stability of a grasp. Most of the works in the literature represented tactile readings as plain feature vectors or matrix-like tactile images, using them to train machine learning models. In this work, we explore an alternative way of exploiting tactile information to predict grasp stability by leveraging graph-like representations of tactile data, which preserve the actual spatial arrangement of the sensor's taxels and their locality. In experimentation, we trained a Graph Neural Network to binary classify grasps as stable or slippery ones. To train such network and prove its predictive capabilities for the problem at hand, we captured a novel dataset of approximately 5000 three-fingered grasps across 41 objects for training and 1000 grasps with 10 unknown objects for testing. Our experiments prove that this novel approach can be effectively used to predict grasp stability