Combining Problems on RAC Drawings and Simultaneous Graph Drawings

We present an overview of the first combinatorial results for the so-called geometric RAC simultaneous drawing problem (or GRacSim drawing problem, for short), i.e., a combination of problems on geometric RAC drawings [3] and geometric simultaneous graph drawings [2]. According to this problem, we are given two planar graphs G1 = (V,E1) and G2 = (V,E2) that share a common vertex set but have disjoint edge sets, i.e., E1 ⊆ V × V , E2 ⊆ V × V and E1 ∩E2 = ∅. The main task is to place the vertices on the plane so that, when the edges are drawn as straight-lines, (i) each graph is drawn planar, (ii) there are no edge overlaps, and, (iii) crossings between edges in E1 and E2 occur at right angles. A closely related problem is the following: Given a planar embedded graph G, determine a geometric drawing of G and its dual G∗ (without the face-vertex corresponding to the external face) such that: (i) G and G∗ are drawn planar, (ii) each vertex of the dual is drawn inside its corresponding face of G and, (iii) the primal-dual edge crossings form right-angles. We refer to this problem as the geometric Graph-Dual RAC simultaneous drawing problem (or GDualGRacSim for short).