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

Drawing Trees with Perfect Angular Resolution and Polynomial Area

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2{\pi}/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.Comment: 30 pages, 17 figure

Persistent Homology Guided Force-Directed Graph Layouts

Graphs are commonly used to encode relationships among entities, yet their abstractness makes them difficult to analyze. Node-link diagrams are popular for drawing graphs, and force-directed layouts provide a flexible method for node arrangements that use local relationships in an attempt to reveal the global shape of the graph. However, clutter and overlap of unrelated structures can lead to confusing graph visualizations. This paper leverages the persistent homology features of an undirected graph as derived information for interactive manipulation of force-directed layouts. We first discuss how to efficiently extract 0-dimensional persistent homology features from both weighted and unweighted undirected graphs. We then introduce the interactive persistence barcode used to manipulate the force-directed graph layout. In particular, the user adds and removes contracting and repulsing forces generated by the persistent homology features, eventually selecting the set of persistent homology features that most improve the layout. Finally, we demonstrate the utility of our approach across a variety of synthetic and real datasets

An Iterative and Toolchain-Based Approach to Automate Scanning and Mapping Computer Networks

As today's organizational computer networks are ever evolving and becoming more and more complex, finding potential vulnerabilities and conducting security audits has become a crucial element in securing these networks. The first step in auditing a network is reconnaissance by mapping it to get a comprehensive overview over its structure. The growing complexity, however, makes this task increasingly effortful, even more as mapping (instead of plain scanning), presently, still involves a lot of manual work. Therefore, the concept proposed in this paper automates the scanning and mapping of unknown and non-cooperative computer networks in order to find security weaknesses or verify access controls. It further helps to conduct audits by allowing comparing documented with actual networks and finding unauthorized network devices, as well as evaluating access control methods by conducting delta scans. It uses a novel approach of augmenting data from iteratively chained existing scanning tools with context, using genuine analytics modules to allow assessing a network's topology instead of just generating a list of scanned devices. It further contains a visualization model that provides a clear, lucid topology map and a special graph for comparative analysis. The goal is to provide maximum insight with a minimum of a priori knowledge.Comment: 7 pages, 6 figure

Graph Algorithm Animation with Grrr

We discuss geometric positioning, highlighting of visited nodes and user defined highlighting that form the algorithm animation facilities in the Grrr graph rewriting programming language. The main purpose of animation was initially for the debugging and profiling of Grrr code, but recently it has been extended for the purpose of teaching algorithms to undergraduate students. The animation is restricted to graph based algorithms such as graph drawing, list manipulation or more traditional graph theory. The visual nature of the Grrr system allows much animation to be gained for free, with no extra user effort beyond the coding of the algorithm, but we also discuss user defined animations, where custom algorithm visualisations can be explicitly defined for teaching and demonstration purposes

Load-Balancing for Parallel Delaunay Triangulations

Computing the Delaunay triangulation (DT) of a given point set in $\mathbb{R}^D$ is one of the fundamental operations in computational geometry. Recently, Funke and Sanders (2017) presented a divide-and-conquer DT algorithm that merges two partial triangulations by re-triangulating a small subset of their vertices - the border vertices - and combining the three triangulations efficiently via parallel hash table lookups. The input point division should therefore yield roughly equal-sized partitions for good load-balancing and also result in a small number of border vertices for fast merging. In this paper, we present a novel divide-step based on partitioning the triangulation of a small sample of the input points. In experiments on synthetic and real-world data sets, we achieve nearly perfectly balanced partitions and small border triangulations. This almost cuts running time in half compared to non-data-sensitive division schemes on inputs exhibiting an exploitable underlying structure.Comment: Short version submitted to EuroPar 201

(D+1)(D+1)-Colored Graphs - a Review of Sundry Properties

We review the combinatorial, topological, algebraic and metric properties supported by $(D+1)$-colored graphs, with a focus on those that are pertinent to the study of tensor model theories. We show how to extract a limiting continuum metric space from this set of graphs and detail properties of this limit through the calculation of exponents at criticality