Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

When can a plane graph with prescribed edge lengths and prescribed angles (from among $\{0,180^\circ, 360^\circ$\}) be folded flat to lie in an infinitesimally thin line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^\circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure

Multilevel Planarity

In this paper, we introduce and study multilevel planarity, a generalization of upward planarity and level planarity. Let $G = (V, E)$ be a directed graph and let $\ell: V \to \mathcal P(\mathbb Z)$ be a function that assigns a finite set of integers to each vertex. A multilevel-planar drawing of $G$ is a planar drawing of $G$ such that for each vertex $v\in V$ its $y$-coordinate $y(v)$ is in $\ell(v)$, nd each edge is drawn as a strictly $y$-monotone curve. We present linear-time algorithms for testing multilevel planarity of embedded graphs with a single source and of oriented cycles. Complementing these algorithmic results, we show that multilevel-planarity testing is NP-complete even in very restricted cases

Constrained Planarity in Practice -- Engineering the Synchronized Planarity Algorithm

In the constrained planarity setting, we ask whether a graph admits a planar drawing that additionally satisfies a given set of constraints. These constraints are often derived from very natural problems; prominent examples are Level Planarity, where vertices have to lie on given horizontal lines indicating a hierarchy, and Clustered Planarity, where we additionally draw the boundaries of clusters which recursively group the vertices in a crossing-free manner. Despite receiving significant amount of attention and substantial theoretical progress on these problems, only very few of the found solutions have been put into practice and evaluated experimentally. In this paper, we describe our implementation of the recent quadratic-time algorithm by Bl\"asius et al. [TALG Vol 19, No 4] for solving the problem Synchronized Planarity, which can be seen as a common generalization of several constrained planarity problems, including the aforementioned ones. Our experimental evaluation on an existing benchmark set shows that even our baseline implementation outperforms all competitors by at least an order of magnitude. We systematically investigate the degrees of freedom in the implementation of the Synchronized Planarity algorithm for larger instances and propose several modifications that further improve the performance. Altogether, this allows us to solve instances with up to 100 vertices in milliseconds and instances with up to 100 000 vertices within a few minutes.Comment: to appear in Proceedings of ALENEX 202

Practical Level Planarity Testing and Layout with Embedding Constraints

We describe a practical method to test a leveled graph for level planarity and provide a level planar layout of the graph if the test succeeds, all in quadratic running-time. Embedding constraints restricting the order of incident edges around the vertices are allowed

Zyklische Levelzeichnungen gerichteter Graphen

The Sugiyama framework proposed in the seminal paper of 1981 is one of the most important algorithms in graph drawing and is widely used for visualizing directed graphs. In its common version, it draws graphs hierarchically and, hence, maps the topological direction to a geometric direction. However, such a hierarchical layout is not possible if the graph contains cycles, which have to be destroyed in a preceding step. In certain application and problem settings, e.g., bio sciences or periodic scheduling problems, it is important that the cyclic structure of the input graph is preserved and clearly visible in drawings. Sugiyama et al. also suggested apart from the nowadays standard horizontal algorithm a cyclic version they called recurrent hierarchies. However, this cyclic drawing style has not received much attention since. In this thesis we consider such cyclic drawings and investigate the Sugiyama framework for this new scenario. As our goal is to visualize cycles directly, the first phase of the Sugiyama framework, which is concerned with removing such cycles, can be neglected. The cyclic structure of the graph leads to new problems in the remaining phases, however, for which solutions are proposed in this thesis. The aim is a complete adaption of the Sugiyama framework for cyclic drawings. To complement our adaption of the Sugiyama framework, we also treat the problem of cyclic level planarity and present a linear time cyclic level planarity testing and embedding algorithm for strongly connected graphs