Testing the Simultaneous Embeddability of Two Graphs Whose Intersection Is a Biconnected Graph or a Tree

In this paper we study the time complexity of the problem Simultaneous Embedding with Fixed Edges (SEFE), that takes two planar graphs G(1) = (V, E-1) and G(2) = (V, E-2) as input and asks whether a planar drawing Gamma(1) of G(1) and a planar drawing 12 of G2 exist such that: (i) each vertex v epsilon V is mapped to the same point in Gamma(1), and in Gamma(2); (ii) every edge e epsilon E-1 boolean AND E-2 is mapped to the same Jordan curve in Gamma(1) and Gamma(2). First, we show a polynomial-time algorithm for SEFE when the intersection graph of G(1) and G(2), that is the planar graph G(1 boolean AND 2) = (V, E-1 boolean AND E-2). is biconnected. Second, we show that SEFE, when G(1 boolean AND 2) is a tree, is equivalent to a suitably-defined hook 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 G(1 boolean AND 2) is a star.