Simultaneous embedding of embedded planar graphs

Abstract

Given k planar graphs G 1,...,G k , deciding whether they admit a simultaneous embedding with fixed edges (Sefe) and whether they admit a simultaneous geometric embedding (Sge ) are NP-hard problems, for k ≥ 3 and for k ≥ 2, respectively. In this paper we consider the complexity of Sefe and of Sge when the graphs G 1,...,G k have a fixed planar embedding. In sharp contrast with the NP-hardness of Sefe for three non-embedded graphs, we show that Sefe is polynomial-time solvable for three graphs with a fixed planar embedding. Furthermore, we show that, given k embedded planar graphs G 1,...,G k , deciding whether a Sefe of G 1,...,G k exists and deciding whether an Sge of G 1,...,G k exists are NP-hard problems, for k ≥ 14 and k ≥ 13, respectively. © 2011 Springer-Verlag.