Simultaneous embedding of embedded planar graphs

Authors:
Patrizio Angelini;Giuseppe Di Battista;Fabrizio Frati
Affiliations:
Dipartimento di Informatica e Automazione, Università Roma Tre, Italy;Dipartimento di Informatica e Automazione, Università Roma Tre, Italy;Dipartimento di Informatica e Automazione, Università Roma Tre, Italy
Venue:
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Year:
2011

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Abstract

Given k planar graphs G1,…,Gk, 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 G1,…,Gk 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 G1,…,Gk, deciding whether a Sefe of G1,…,Gk exists and deciding whether an Sge of G1,…,Gk exists are NP-hard problems, for k≥14 and k≥13, respectively.