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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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2013

## Authors and Affiliations

• Thomas Bläsius
• 1
• Ignaz Rutter
• 1
1. 1.Karlsruhe Institute of Technology (KIT)Germany

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