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