Simultaneous embedding of outerplanar graphs, paths, and cycles

Let G(1) and G(2) be two planar graphs having some vertices in common. A simulteneous embedding of G(1) and G(2) is a pair of crossing-free drawings of G(1) and G(2) vertex in common is represented by the same point in both drawings. In this paper we show that an outerplanar graph and a simple path can be simultaneously embedded with fixed edges such that athe edges din common are straight-line segments while the other edges of the outerplanar graph can have at most one bend per edge. We then exploit the technique for outerplanar graphs and paths to study simultaneous embeddings of other pairs of graphs. Namely, we study simultaneous embeeding with fixed edges of: (i) two outerplanar graphs sharing a forest of paths and (ii) an outerplanar graph and a cycle.