Geometric RAC Simultaneous Drawings of Graphs

Abstract

In this paper, we introduce and study geometric simultaneous RAC drawing problems, i.e., a combination of problems on geometric RAC drawings and geometric simultaneous graph drawings. To the best of our knowledge, this is the first time where such a combination is attempted.

Simultaneous visibility representations of plane st-graphs using L-shapes

“…Given two planar graphs with the same vertex set, an SRE is a simultaneous embedding where crossings between edges of the two graphs occur at right angles. Argyriou et al proved that it is always possible to construct an SRE with straight-line edges of a cycle and a matching, while there exist a wheel graph and a cycle that do not admit such a representation [2]. This motivated recent results about SRE with bends along the edges.…”

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].…”

“…In other related work, it is shown in that any ‐colorable graph has a grid drawing with area in which no vertices of are interior to any edges. 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 . In this paper, we are interested in embeddings of the following form.…”

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

“…In this paper we study the complexity of the GEOMETRIC RAC SIMULTANEOUS DRAWING problem [8] (GRACSIM DRAWING for short): a restricted version of SGE, which asks for finding a simultaneous geometric embedding of two graphs, such that all edge crossings must occur at right angles. We show that GRACSIM DRAWING is N P-hard by a reduction from 3-PARTITION; see Section 3.…”