Testing the simultaneous embeddability of two graphs whose intersection is a biconnected graph or a tree

Authors:
Patrizio Angelini;Giuseppe Di Battista;Fabrizio Frati;Maurizio Patrignani;Ignaz Rutter
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;Dipartimento di Informatica e Automazione, Università Roma Tre, Italy;Institute of Theoretical Informatics, Karlsruhe Institute of Technology, Germany
Venue:
IWOCA'10 Proceedings of the 21st international conference on Combinatorial algorithms
Year:
2010

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Abstract

In this paper we study the time complexity of the problem Simultaneous Embedding with Fixed Edges (SEFE), that takes two planar graphs G1 = (V, E1) and G2 = (V, E2) as input and asks whether a planar drawing Γ1 of G1 and a planar drawing Γ2 of G2 exist such that: (i) each vertex v ∈ V is mapped to the same point in Γ1 and in Γ2; (ii) every edge e ∈ E1 ∩ E2 is mapped to the same Jordan curve in Γ1 and Γ2. First, we show a polynomial-time algorithm for SEFE when the intersection graph of G1 and G2, that is the planar graph G1∩2 = (V, E1 ∩ E2), is biconnected. Second, we show that SEFE, when G1∩2 is a tree, is equivalent to a suitably-defined book embedding problem. Based on such an equivalence and on recent results by Hong and Nagamochi, we show a linear-time algorithm for the SEFE problem when G1∩2 is a star.