Abstract We say that a (multi)graph $$ \user2{G} = (\user2{V},\user2{E}) $$ has geometric thickness t if there exists a straight-line drawing $$ \user2{\varphi }:\user2{V} \to \mathbb{R}^{{\mathbf{2}}} $$ and a t -coloring of its edges where no two edges sharing a point in their relative interior have the same color. The Geometric Thickness problem asks whether a given multigraph has geometric thickness at most t . This problem was shown to be NP-hard for $$ \user2{t} = \mathbf{2} $$ (Durocher et al. Comput Geom 56:1–18, 2016. https://doi.org/10.1016/j.comgeo.2016.03.003 ). In this paper, we settle the computational complexity of Geometric Thickness by showing that it is $$\exists \mathbb {R}$$ -complete already for thickness 30 . Moreover, our reduction shows that the problem is $$\exists \mathbb {R}$$ -complete for 4392 -planar graphs, where a graph is k -planar if it admits a topological drawing with at most k crossings per edge. In the course of our paper we answer previous questions on geometric thickness and on other related problems, in particular that simultaneous graph embeddings of 31 edge-disjoint graphs and pseudo-segment stretchability with chromatic number 30 are $$\exists \mathbb {R}$$ -complete.

London's famous "tube map" is an iconic piece of design and perhaps represents the schematic visualization style most well-known to the general public: its octolinearity has become the de facto standard for transit maps around the world. Making a good schematic transit map is challenging and labour-intensive, and has attracted the attention of the optimization community. Much of the literature has focused on mathematically defining an optimal drawing and algorithms to compute one. However, achieving these "optimal" layouts is computationally challenging, often requiring multiple minutes of runtime. Crucially, what it means for a map to be good is actually highly dependent on factors that evade a general formal definition, like unique landmarks within the network, the context in which a map will be displayed, and the preference of the designer and client. Rather than attempting to make an algorithm that produces a single high-quality and ready-to-use metro map at great cost, we propose it is more fruitful to support rapid layout iteration by a human designer, providing a workflow that enables efficient exploration of a wider range of designs than could be done by hand, and iterating on these designs. To this end we identify steps in the design process of schematic maps that are tedious to do by hand but are algorithmically feasible, and present a framework around a simple linear program that computes network layouts almost instantaneously given a fixed direction for every connection. These connection directions are decided by a designer in a graphical user interface with several interaction methods and a number of quality-of-life features demonstrating the flexibility of the framework; the implementation is available as open source.

A wUSR of a graph G is a set of unit squares in the plane, one per vertex, if two vertices have an edge in G if their squares overlap and the overlap contains no witness. We present an output sensitive algorithm to compute a graph G based on its given witness unit square representation.

Boundary labeling is a technique in computational geometry used to label sets of features in an illustration. It involves placing labels along an axis-parallel bounding box and connecting each label with its corresponding feature using non-crossing leader lines. Although boundary labeling is well-studied, semantic constraints on the labels have not been investigated thoroughly. In this paper, we introduce grouping and ordering constraints in boundary labeling: Grouping constraints enforce that all labels in a group are placed consecutively on the boundary, and ordering constraints enforce a partial order over the labels. We show that it is NP -hard to find a labeling for arbitrarily sized labels with unrestricted positions along one side of the boundary. However, we obtain polynomial-time algorithms if we restrict this problem either to uniform-height labels or to a finite set of candidate positions. Furthermore, we show that finding a labeling on two opposite sides of the boundary is NP -complete, even for uniform-height labels and finite label positions. Finally, we experimentally confirm that our approach has also practical relevance.

Boundary labeling is a well-known method for displaying short textual labels for a set of point features in a figure alongside the boundary of that figure. Labels and their corresponding points are connected via crossing-free leaders. We propose orbital boundary labeling as a new variant of the problem, in which (i) the figure is enclosed by a circular contour and (ii) the labels are placed as disjoint circular arcs in an annulus-shaped orbit around the contour. The algorithmic objective is to compute an orbital boundary labeling with the minimum total leader length. We identify several parameters that define the corresponding problem space: two leader types (straight or orbital-radial), label size and order, presence of candidate label positions, and constraints on where a leader attaches to its label. Our results provide polynomial-time algorithms for many variants and NP-hardness for others, using a variety of geometric and combinatorial insights.

Boundary labeling is a technique in computational geometry used to label sets of features in an illustration. It involves placing labels along an axis-parallel bounding box and connecting each label with its corresponding feature using non-crossing leader lines. Although boundary labeling is well-studied, semantic constraints on the labels have not been investigated thoroughly. In this paper, we introduce grouping and ordering constraints in boundary labeling: Grouping constraints enforce that all labels in a group are placed consecutively on the boundary, and ordering constraints enforce a partial order over the labels. We show that it is NP-hard to find a labeling for arbitrarily sized labels with unrestricted positions along one side of the boundary. However, we obtain polynomial-time algorithms if we restrict this problem either to uniform-height labels or to a finite set of candidate positions. Furthermore, we show that finding a labeling on two opposite sides of the boundary is NP-complete, even for uniform-height labels and finite label positions. Finally, we experimentally confirm that our approach has also practical relevance.

We study a variant of the geometric multicut problem, where we are given a set $\mathcal{P}$ of colored and pairwise interior-disjoint polygons in the plane. The objective is to compute a set of simple closed polygon boundaries (fences) that separate the polygons in such a way that any two polygons that are enclosed by the same fence have the same color, and the total number of links of all fences is minimized. We call this the minimum link fencing (MLF) problem and consider the natural case of bounded minimum link fencing (BMLF), where $\mathcal{P}$ contains a polygon $Q$ that is unbounded in all directions and can be seen as an outer polygon. We show that BMLF is NP-hard in general and that it is XP-time solvable when each fence contains at most two polygons and the number of segments per fence is the parameter. Finally, we present an $O(n \log n)$-time algorithm for the case that the convex hull of $\mathcal{P} \setminus \{Q\}$ does not intersect $Q$.

Metro or transit maps, are schematic representations of transit networks to facilitate effective route-finding. These maps are often advertised on a web page or pamphlet highlighting routes from source to destination stations. To visually support such route-finding, designers often distort the layout by embedding symbolic shapes (e.g., circular routes) in order to guide readers' attention (e.g., Moscow map and Japan railway map). However, manually producing such maps is labor-intensive and the effect of shapes remains unclear. In this paper, we propose an approach to generalize such mixed metro maps that take user-defined shapes as an input. In this mixed design, lines that are used to approximate the shapes are arranged symbolically, while the remaining lines follow classical layout convention. A three-step algorithm, including (1) detecting and selecting routes for shape approximation, (2) shape and layout deformation, and (3) aligning lines on a grid, is integrated to guarantee good visual quality. Our contribution lies in the definition of the mixed metro map problem and the formulation of design criteria so that the problem can be resolved systematically using the optimization paradigm. Finally, we evaluate the performance of our approach and perform a user study to test if the embedded shapes are recognizable or reduce the map quality.

Cartograms are popular for visualizing numerical data for administrative regions in thematic maps. When there are multiple data values per region (over time or from different datasets) shown as animated or juxtaposed cartograms, preserving the viewer's mental map in terms of stability between multiple cartograms is another important criterion alongside traditional cartogram criteria such as maintaining adjacencies. We present a method to compute stable stable Demers cartograms, where each region is shown as a square scaled proportionally to the given numerical data and similar data yield similar cartograms. We enforce orthogonal separation constraints using linear programming, and measure quality in terms of keeping adjacent regions close (cartogram quality) and using similar positions for a region between the different data values (stability). Our method guarantees the ability to connect most lost adjacencies with minimal-length planar orthogonal polylines. Experiments show that our method yields good quality and stability on multiple quality criteria.