GD 2013: Graph Drawing pp 220-231

# Simultaneous Embedding: Edge Orderings, Relative Positions, Cutvertices

• Thomas Bläsius
• Annette Karrer
• Ignaz Rutter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)

## Abstract

A simultaneous embedding of two graphs $$G^{\mbox{\textcircled{1}}}$$ and $$G^{\mbox{\textcircled{2}}}$$ with common graph $$G=G^{\mbox{\textcircled{1}}} \cap G^{\mbox{\textcircled{2}}}$$ is a pair of planar drawings of $$G^{\mbox{\textcircled{1}}}$$ and $$G^{\mbox{\textcircled{2}}}$$ that coincide on G. It is an open question whether there is a polynomial-time algorithm that decides whether two graphs admit a simultaneous embedding (problem Sefe).

In this paper, we present two results. First, a set of three linear-time preprocessing algorithms that remove certain substructures from a given Sefe instance, producing a set of equivalent Sefe instances without such substructures. The structures we can remove are (1) cutvertices of the union graph$$G^{\mbox{\textcircled{1}}} \cup G^{\mbox{\textcircled{2}}}$$, (2) cutvertices that are simultaneously a cutvertex in $$G^{\mbox{\textcircled{1}}}$$ and $$G^{\mbox{\textcircled{2}}}$$ and that have degree at most 3 in G, and (3) connected components of G that are biconnected but not a cycle.

Second, we give an O(n2)-time algorithm for Sefe where, for each pole u of a P-node μ (of a block) of the input graphs, at most three virtual edges of μ contain common edges incident to u. All algorithms extend to the sunflower case.

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© Springer International Publishing Switzerland 2013

## Authors and Affiliations

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