Aligned Drawings of Planar Graphs

Let $G$ be a graph that is topologically embedded in the plane and let $\mathcal{A}$ be an arrangement of pseudolines intersecting the drawing of $G$. An aligned drawing of $G$ and $\mathcal{A}$ is a planar polyline drawing $\Gamma$ of $G$ with an arrangement $A$ of lines so that $\Gamma$ and $A$ are homeomorphic to $G$ and $\mathcal{A}$. We show that if $\mathcal{A}$ is stretchable and every edge $e$ either entirely lies on a pseudoline or it has at most one intersection with $\mathcal{A}$, then $G$ and $\mathcal{A}$ have a straight-line aligned drawing. In order to prove this result, we strengthen a result of Da Lozzo et al., and prove that a planar graph $G$ and a single pseudoline $\mathcal{L}$ have an aligned drawing with a prescribed convex drawing of the outer face. We also study the less restrictive version of the alignment problem with respect to one line, where only a set of vertices is given and we need to determine whether they can be collinear. We show that the problem is NP-complete but fixed-parameter tractable.Comment: Preliminary work appeared in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

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.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Planar L-Drawings of Directed Graphs

We study planar drawings of directed graphs in the L-drawing standard. We provide necessary conditions for the existence of these drawings and show that testing for the existence of a planar L-drawing is an NP-complete problem. Motivated by this result, we focus on upward-planar L-drawings. We show that directed st-graphs admitting an upward- (resp. upward-rightward-) planar L-drawing are exactly those admitting a bitonic (resp. monotonically increasing) st-ordering. We give a linear-time algorithm that computes a bitonic (resp. monotonically increasing) st-ordering of a planar st-graph or reports that there exists none.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Planar Drawings of Fixed-Mobile Bigraphs

A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given k >= 0, a planar poly-line drawing of G with at most k bends per edge. In the most general case, we show NP-hardness. For k=0 and under additional constraints on the positions of the fixed or mobile vertices, we either prove that the problem is polynomial-time solvable or prove that it belongs to NP. Finally, we present a polynomial-time testing algorithm for a certain type of "layered" 1-bend drawings

Solving Rectilinear Steiner Tree Problems Exactly in Theory and Practice

The rectilinear Steiner tree problem asks for a shortest tree connecting given points in the plane with rectilinear distance. The best theoretically analyzed algorithm for this problem with a fairly practical behaviour bases on dynamic programming and has a running time of O(n 2 \Delta 2:62 n ) (Ganley/Cohoon). The best implementation can solve random problems of size 35 (Salowe/Warme) within a day. In this paper we improve the theoretical worst-case time bound to O(n \Delta 2:38 n ), for random problem instances we prove a running time of less than O(2 n ). In practice, our ideas lead to even more drastic improvements. Extensive experiments show that the range for the size of random problems solvable within a day on a workstation is almost doubled. For exponential time algorithms, this is an enormous step

Faster Approximation Algorithms forthe Rectilinear Steiner Tree Problem

Abstract. The classical Steiner tree problem requires a shortest tree spanning a given vertex subset within a graph G = (V, E). An important variant is the Steiner tree problem in rectilinear metric. Only recently two algorithms were found which achieve better approximations than the “traditional ” one with a factor of 3/2. These algorithms with an approximation ratio of 11/8 are quite slow and run in time O(n 3) and O(n 5/2).Anew simple implementation reduces the time to O(n 3/2). As our main result we present efficient parametrized algorithms which reach a performance ratio of 11/8+ε for any ε>0 in time O(n · log 2 n), and a ratio of 11/8 + log log n/log n in time O(n · log 3 n). 1

Dynamic graph drawing of sequences of orthogonal and hierarchical graphs

Abstract. In this paper we introduce two novel algorithms for drawing sequences of orthogonal and hierarchical graphs while preserving the mental map. Both algorithms can be parameterized to trade layout quality for dynamic stability. In particular, we had to develop new metrics which work upon the intermediate results of layout phases. We discuss some properties of the resulting animations by means of examples.

Semi-dynamic Orthogonal Drawings of Planar Graphs (Extended Abstract)

We introduce a new approach to orthogonal drawings of planar graphs. We define invariants that are respected by every drawing of the graph. The invariants are the embedding together with relative positions of adjacent vertices. Insertions imply only minor changes of the invariants. This preserves the users mental map. Our technique is applicable to two-connected planar graphs with vertices of arbitrary size and degree. New vertices and edges can be added to the graph in O(log n) time. The algorithm produces drawings with at most m + f bends, where m and f are the number of edges and faces of the graph

Fast Compaction for Orthogonal Drawings with Vertices of Prescribed Size

In this paper, we present a new compaction algorithm which computes orthogonal drawings where the size of the vertices is given as input. This is a critical constraint for many practical applications like UML. The algorithm provides a drastic improvement on previous approaches. It has linear worst case running time and experiments show that it performs very well in practice