On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings

We study two variants of the well-known orthogonal drawing model: (i) the smooth orthogonal, and (ii) the octilinear. Both models form an extension of the orthogonal, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively). For planar graphs of max-degree 4, we analyze relationships between the graph classes that can be drawn bendless in the two models and we also prove NP-hardness for a restricted version of the bendless drawing problem for both models. For planar graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal drawings with at most two segments per edge, which also guarantees a linear number of edges with exactly one segment

Simple Compact Monotone Tree Drawings.

A monotone drawing of a graph G is a straight-line drawing of G such that every pair of vertices is connected by a path that is monotone with respect to some direction. Trees, as a special class of graphs, have been the focus of several papers and, recently, He and He [6] showed how to produce a monotone drawing of an arbitrary n-vertex tree that is contained in a 12n×12n grid. In this paper, we present a simple algorithm that constructs for each arbitrary tree a monotone drawing on a grid of size at most n×n

Reconstructing Generalized Staircase Polygons with Uniform Step Length

Visibility graph reconstruction, which asks us to construct a polygon that has a given visibility graph, is a fundamental problem with unknown complexity (although visibility graph recognition is known to be in PSPACE). We show that two classes of uniform step length polygons can be reconstructed efficiently by finding and removing rectangles formed between consecutive convex boundary vertices called tabs. In particular, we give an O(n2m)-time reconstruction algorithm for orthogonally convex polygons, where n and m are the number of vertices and edges in the visibility graph, respectively. We further show that reconstructing a monotone chain of staircases (a histogram) is fixed-parameter tractable, when parameterized on the number of tabs, and polynomially solvable in time O(n2m) under reasonable alignment restrictions

Optimal Compaction of Orthogonal Grid Drawings for Graphs

Orthogonal graphs are used in a multitude of applications to visualize information. Examples include database design, software engineering, VLSI layout and UML diagrams. The TSM approach is an effective methodology for creating orthogonal grid drawings of graphs. Its name is an acronym of its three stages: topology, in which a planar representation is defined; shape, when an orthogonal representation is obtained; and metrics, in which the graph’s elements are positioned on the grid in accordance to the orthogonal representation, while optimizing some characteristic of the drawing

EPG-representations with Small Grid-Size

In an EPG-representation of a graph G, each vertex is represented by a path in the rectangular grid, and (v, w) is an edge in G if and only if the paths representing v an w share a grid-edge. Requiring paths representing edges to be x-monotone or, even stronger, both x- and y-monotone gives rise to three natural variants of EPG-representations, one where edges have no monotonicity requirements and two with the aforementioned monotonicity requirements. The focus of this paper is understanding how small a grid can be achieved for such EPG-representations with respect to various graph parameters. We show that there are m-edge graphs that require a grid of area Ω(m) in any variant of EPG-representations. Similarly there are pathwidth-k graphs that require height Ω(k) and area Ω(kn) in any variant of EPG-representations. We prove a matching upper bound of O(kn) area for all pathwidth-k graphs in the strongest model, the one where edges are required to be both x- and y-monotone. Thus in this strongest model, the result implies, for example, O(n), O(nlogn) and O(n^3/2) area bounds for bounded pathwidth graphs, bounded treewidth graphs and all classes of graphs that exclude a fixed minor, respectively. For the model with no restrictions on the monotonicity of the edges, stronger results can be achieved for some graph classes, for example an O(n) area bound for bounded treewidth graphs and O(nlog^2 n) bound for graphs of bounded genus

The Effect of Planarization on Width.

We study the effects of planarization (the construction of a planar diagram D from a non-planar graph G by replacing each crossing by a new vertex) on graph width parameters. We show that for treewidth, pathwidth, branchwidth, clique-width, and tree-depth there exists a family of n-vertex graphs with bounded parameter value, all of whose planarizations have parameter value Ω(n). However, for bandwidth, cutwidth, and carving width, every graph with bounded parameter value has a planarization of linear size whose parameter value remains bounded. The same is true for the treewidth, pathwidth, and branchwidth of graphs of bounded degree

Computing Storyline Visualizations with Few Block Crossings

Storyline visualizations show the structure of a story, by depicting the interactions of the characters over time. Each character is represented by an x-monotone curve from left to right, and a meeting is represented by having the curves of the participating characters run close together for some time. There have been various approaches to drawing storyline visualizations in an automated way. In order to keep the visual complexity low, rather than minimizing pairwise crossings of curves, we count block crossings, that is, pairs of intersecting bundles of lines. Partly inspired by the ILP-based approach of Gronemann et al. [GD 2016] for minimizing the number of pairwise crossings, we model the problem as a satisfiability problem (since the straightforward ILP formulation becomes more complicated and harder to solve). Having restricted ourselves to a decision problem, we can apply powerful SAT solvers to find optimal drawings in reasonable time. We compare this SAT-based approach with two exact algorithms for block crossing minimization, using both the benchmark instances of Gronemann et al. and random instances. We show that the SAT approach is suitable for real-world instances and identify cases where the other algorithms are preferable

Thrackles: An Improved Upper Bound.

A thrackle is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of n vertices has at most 1.3984n edges. Quasi-thrackles are defined similarly, except that every pair of edges that do not share a vertex are allowed to cross an odd number of times. It is also shown that the maximum number of edges of a quasi-thrackle on n vertices is 3/2(n−1), and that this bound is best possible for infinitely many values of n