Planar Octilinear Drawings with One Bend Per Edge

In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal ($45^\circ$) line-segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A $k$-planar graph is a planar graph in which each vertex has degree less or equal to $k$. In particular, we prove that every 4-planar graph admits a planar octilinear drawing with at most one bend per edge on an integer grid of size $O(n^2) \times O(n)$. For 5-planar graphs, we prove that one bend per edge still suffices in order to construct planar octilinear drawings, but in super-polynomial area. However, for 6-planar graphs we give a class of graphs whose planar octilinear drawings require at least two bends per edge

Relaxing the Constraints of Clustered Planarity

In a drawing of a clustered graph vertices and edges are drawn as points and curves, respectively, while clusters are represented by simple closed regions. A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region, or region-region crossings. Determining the complexity of testing whether a clustered graph admits a c-planar drawing is a long-standing open problem in the Graph Drawing research area. An obvious necessary condition for c-planarity is the planarity of the graph underlying the clustered graph. However, such a condition is not sufficient and the consequences on the problem due to the requirement of not having edge-region and region-region crossings are not yet fully understood. In order to shed light on the c-planarity problem, we consider a relaxed version of it, where some kinds of crossings (either edge-edge, edge-region, or region-region) are allowed even if the underlying graph is planar. We investigate the relationships among the minimum number of edge-edge, edge-region, and region-region crossings for drawings of the same clustered graph. Also, we consider drawings in which only crossings of one kind are admitted. In this setting, we prove that drawings with only edge-edge or with only edge-region crossings always exist, while drawings with only region-region crossings may not. Further, we provide upper and lower bounds for the number of such crossings. Finally, we give a polynomial-time algorithm to test whether a drawing with only region-region crossings exist for biconnected graphs, hence identifying a first non-trivial necessary condition for c-planarity that can be tested in polynomial time for a noticeable class of graphs

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

On Compact RAC Drawings

We present new bounds for the required area of Right Angle Crossing (RAC) drawings for complete graphs, i.e. drawings where any two crossing edges are perpendicular to each other. First, we improve upon results by Didimo et al. [Walter Didimo et al., 2011] and Di Giacomo et al. [Emilio Di Giacomo et al., 2011] by showing how to compute a RAC drawing with three bends per edge in cubic area. We also show that quadratic area can be achieved when allowing eight bends per edge in general or with three bends per edge for p-partite graphs. As a counterpart, we prove that in general quadratic area is not sufficient for RAC drawings with three bends per edge

L-Visibility Drawings of IC-planar Graphs

An IC-plane graph is a topological graph where every edge is crossed at most once and no two crossed edges share a vertex. We show that every IC-plane graph has a visibility drawing where every vertex is an L-shape, and every edge is either a horizontal or vertical segment. As a byproduct of our drawing technique, we prove that an IC-plane graph has a RAC drawing in quadratic area with at most two bends per edge

Edge Routing with Ordered Bundles

Edge bundling reduces the visual clutter in a drawing of a graph by uniting the edges into bundles. We propose a method of edge bundling drawing each edge of a bundle separately as in metro-maps and call our method ordered bundles. To produce aesthetically looking edge routes it minimizes a cost function on the edges. The cost function depends on the ink, required to draw the edges, the edge lengths, widths and separations. The cost also penalizes for too many edges passing through narrow channels by using the constrained Delaunay triangulation. The method avoids unnecessary edge-node and edge-edge crossings. To draw edges with the minimal number of crossings and separately within the same bundle we develop an efficient algorithm solving a variant of the metro-line crossing minimization problem. In general, the method creates clear and smooth edge routes giving an overview of the global graph structure, while still drawing each edge separately and thus enabling local analysis