Disconnectivity and relative positions in simultaneous embeddings

Abstract

For two planar graph GCircle digit 1 = (V Circle digit 1, ECircle digit 1 ) and G Circle digit 2 = (VCircle digit 2 ,E Circle digit 2) sharing a common subgraph G = G Circle digit 1 ∩ GCircle digit 2 the problem Simultaneous Embedding with Fixed Edges (SEFE) asks whether they admit planar drawings such that the common graph is drawn the same. Previous algorithms only work for cases where G is connected, and hence do not need to handle relative positions of connected components. We consider the problem where G, G Circle digit 1 and GCircle digit 2 are not necessarily connected. First, we show that a general instance of SEFE can be reduced in linear time to an equivalent instance where VCircle digit 1 = V Circle digit 2 and GCircle digit 1 and G Circle digit 2 are connected. Second, for the case where G consists of disjoint cycles, we introduce the CC-tree which represents all embeddings of G that extend to planar embeddings of GCircle digit 1. We show that CC-trees can be computed in linear time, and that their intersection is again a CC-tree. This yields a linear-time algorithm for SEFE if all k input graphs (possibly k > 2) pairwise share the same set of disjoint cycles. These results, including the CC-tree, extend to the case where G consists of arbitrary connected components, each with a fixed embedding. Then the running time is O(n 2). © 2013 Springer-Verlag.