Article ID Journal Published Year Pages File Type
431649 Journal of Discrete Algorithms 2012 23 Pages PDF
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)G1=(V,E1) and G2=(V,E2)G2=(V,E2) as input and asks whether a planar drawing Γ1Γ1 of G1G1 and a planar drawing Γ2Γ2 of G2G2 exist such that: (i) each vertex v∈Vv∈V is mapped to the same point in Γ1Γ1 and in Γ2Γ2; (ii) every edge e∈E1∩E2e∈E1∩E2 is mapped to the same Jordan curve in Γ1Γ1 and Γ2Γ2.First, we give a linear-time algorithm for Sefe when the intersection graph   of G1G1 and G2G2, that is the planar graph G1∩2=(V,E1∩E2)G1∩2=(V,E1∩E2), is biconnected. Second, we show that Sefe, when G1∩2G1∩2 is connected, is equivalent to a suitably-defined book embedding problem. Based on this equivalence and on recent results by Hong and Nagamochi, we show a linear-time algorithm for the Sefe problem when G1∩2G1∩2 is a star.

Related Topics
Physical Sciences and Engineering Computer Science Computational Theory and Mathematics
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