Simultaneous Drawing of Planar Graphs with Right-Angle Crossings and Few Bends

Abstract

Abstract. Given two planar graphs that are defined on the same set of vertices, a RAC simultaneous drawing is a drawing of the two graphs where each graph is drawn planar, no two edges overlap, and edges of one graph can cross edges of the other graph only at right angles. In the geometric version of the problem, vertices are drawn as points and edges as straight-line segments. It is known, however, that even pairs of very simple classes of planar graphs (such as wheels and matchings) do not always admit a geometric RAC simultaneous drawing. In order to enlarge the class of graphs that admit RAC simultaneous drawings, we allow edges to have bends. We prove that any pair of planar graphs admits a RAC simultaneous drawing with at most six bends per edge. For more restricted classes of planar graphs (e.g., matchings, paths, cycles, outerplanar graphs, and subhamiltonian graphs), we significantly reduce the required number of bends per edge. All our drawings use quadratic area.

On the NP-hardness of GRacSim drawing and k-SEFE Problems

“…Our results raise two main questions. First, as already mentioned at the end of Section 3, it would be interesting to study the complexity of a relaxed version of the GRAC-SIM DRAWING problem, where a prescribed number of bends per edge are allowed; this open problem was already posed in [9]. In particular, it is not clear whether the reduction given in the proof of Theorem 1 can be adapted for proving the N P-hardness of the one bend extension of GRACSIM.…”

“…The problem of computing a simultaneous embedding of two or more graphs has been extensively explored by the graph drawing community. Indeed, besides its inherent theoretical interest [1,2,4,5,6,7,9,10,11,12,13,14,15,16,17,18,19,22,23,24,25,26], it has several applications in dynamic network visualization, especially when a visual analysis of an evolving network is needed. Although many variants of this problem have been investigated so far, a general formulation for two graphs can be stated as follows: Let G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ) be two planar graphs sharing a common (or shared) subgraph G = (V, E), where V = V 1 ∩ V 2 and E = E 1 ∩ E 2 .…”

Sequentially embeddable graphs

“…any 3-connected planar graph and its dual can be simultaneously embedded with straight line embeddings in a (2n − 2) × (2n − 2) grid [5]. Moreover, there is a recent trend in studying when two planar graphs with the same vertex set admit geometric RAC simultaneous drawings, that is, simultaneous straight-line embeddings in which edges from different graphs intersect at right angles [3,4].…”

Simultaneous Orthogonal Planarity

“…For planar straight-line drawings, the simultaneous embedding problem is called Simultaneous Geometric Embedding and it is known to be NP-hard even for two graphs [17]. Besides simultaneous intersection representation for, e.g., interval graphs [20,14] and permutation and chordal graphs [21], it is only recently that the simultaneous embedding paradigm has been applied to other fundamental planarity-related drawing styles, namely simultaneous level planar drawings [2] and RAC drawings [4,8].…”