In radial drawings of hierarchical graphs the vertices are placed on concentric circles rather than on horizontal lines and the edges are drawn as outwards monotone segments of spirals rather than straight lines as it is both done in the standard Sugiyama framework. This drawing style is well suited for the visualisation of centrality in social networks and similar concepts. Radial drawings also allow a more flexible edge routing than horizontal drawings, as edges can be routed around the center in two directions. In experimental results this reduces the number of crossings by approximately 30 percent on average. Few crossings are one of the major criteria for human readability. This paper is a detailed description of a complete framework for visualizing hierarchical information in a new radial fashion. Particularly, we briefly cover extensions of the level assignment step to benefit by the increasing perimeters of the circles, present three heuristics for crossing reduction in radial level drawings, and also show how to visualize the results.

Consider a graph G drawn in the plane so that each vertex lies on a distinct horizontal line ℓj = {(x, j) | x ∈ R}. The bijection φ that maps the set of n vertices V to a set of distinct horizontal lines ℓj forms a labeling of the vertices. Such a graph G with the labeling φ is called an n-level graph and is said to be n-level planar if it can be drawn with straight-line edges and no crossings while keeping each vertex on its own level. In this paper, we consider the class of trees that are n-level planar regardless of their labeling. We call such trees unlabeled level planar (ULP). Our contributions are three-fold. First, we provide a complete characterization of ULP trees in terms of a pair of forbidden subtrees. Second, we show how to draw ULP trees in linear time. Third, we provide a linear time recognition algorithm for ULP trees.

Abstract. We revisit monotone planar circuits MPCVP, with special attention to circuits with cylindrical embeddings. MPCVP is known to be in NC 3 in general, and in LogDCFL for the special case of upward stratified circuits. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward planarity, and make the characterization partially constructive. We use this construction, and four key reduction lemmas, to obtain several improvements. We show that monotone circuits with embeddings that are stratified cylindrical, cylindrical, planar one-input-face and focused can be evaluated in LogDCFL, AC 1 (LogDCFL), LogCFL and AC 1 (LogDCFL) respectively. We note that the NC 3 algorithm for general MPCVP is in AC 1 (LogCFL) =SAC 2.Finally, we show that monotone circuits with toroidal embeddings can, given such an embedding, be evaluated in NC. 1

Abstract. We present a simple linear time algorithm for drawing level graphs with a given ordering of the vertices within each level. The algorithm draws in a radial fashion without changing the vertex ordering, and therefore without introducing new edge crossings. Edges are drawn as sequences of spiral segments with at most two bends. 1

A level graph is a directed acyclic graph with a level assignment for each node. Such graphs play a prominent role in graph drawing. They express strict dependencies and occur in many areas, e.g., in scheduling problems and program inheritance structures. In this paper we extend level graphs to cyclic level graphs. Such graphs occur as repeating processes in cyclic scheduling, visual data mining, life sciences, and VLSI. We provide a complete study of strongly connected cyclic level graphs. In particular, we present a linear time algorithm for the planarity testing and embedding problem, and we characterize forbidden subgraphs. Our results generalize earlier work on level graphs.

We consider the problem of simultaneous embedding of planar graphs. We demonstrate how to simultaneously embed a path and an n-level planar graph and how to use radial embeddings for curvilinear simultaneous embeddings of a path and an outerplanar graph. We also show how to use star-shaped levels to find 2-bends per path edge simultaneous embeddings of a path and an outerplanar graph. All embedding algorithms run in O(n) time.

A track graph is a graph with its vertex set partitioned into horizontal levels. It is track planar if there are permutations of the vertices on each level such that all edges can be drawn as weak monotone curves without crossings. The novelty and generalisation over level planar graphs is that horizontal edges connecting consecutive vertices on the same level are allowed. We show that track planarity can be reduced to level planarity in linear time. Hence, there are time algorithms for the track planarity test and for the computation of a track planar embedding.

Abstract. In radial drawings of hierarchical graphs the vertices are drawn on concentric circles instead of on horizontal lines as in the standard Sugiyama framework. This drawing style is well suited for the visualisation of centrality in social networks and similar concepts. Radial drawings also allow a more flexible edge routing than horizontal drawings, as edges can be routed around the center in two directions. In experimental results this reduces the number of crossings by approx. 30% on average. This paper is the last step to complete the framework for drawing hierarchical graphs in a radial fashion. We present three heuristics for crossing reduction in radial level drawings of hierarchical graphs, and also briefly cover extensions of the level assignment step to take advantage of the increasing perimeter of the circles. 1

We describe an algorithm for radial layout of undirected graphs, in which nodes are constrained to concentric circles centered at the origin. Such constraints are typical, e.g., in the layout of social networks, when structural centrality is mapped to geometric centrality or when the primary intention of the layout is the display of the vicinity of a distinguished node. Our approach is based on an extension of stress minimization with a weighting scheme that gradually imposes radial constraints on the intermediate layout during the majorization process, and thus is an attempt to preserve as much information about the graph structure as possible. 1

Abstract. In this paper we introduce cyclic level planar graphs, which are a planar version of the recurrent hierarchies from Sugiyama et al. [8] and the cyclic extension of level planar graphs, where the first level is the successor of the last level. We study the testing and embedding problem and solve it for strongly connected graphs in time O(|V | log |V |). 1