GD 2012: Graph Drawing pp 31-42

# Disconnectivity and Relative Positions in Simultaneous Embeddings

• Thomas Bläsius
• Ignaz Rutter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)

## Abstract

For two planar graph $$G^{\textcircled1}$$ = ($$V^{\textcircled1}$$, $$E^{\textcircled1}$$) and $$G^{\textcircled2}$$ = ($$V^{\textcircled2}$$, $$E^{\textcircled2}$$) sharing a common subgraph G = $$G^{\textcircled1}$$$$G^{\textcircled2}$$ 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, $$G^{\textcircled1}$$ and $$G^{\textcircled2}$$ are not necessarily connected.

First, we show that a general instance of SEFE can be reduced in linear time to an equivalent instance where $$V^{\textcircled1}$$ = $$V^{\textcircled2}$$ and $$G^{\textcircled1}$$ and $$G^{\textcircled2}$$ 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 $$G^{\textcircled1}$$. 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).

## Keywords

Planar Graph Input Graph Outer Face Disjoint Cycle Expansion Graph

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