An SPQR-Tree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges

Authors:
J. Joseph Fowler;Carsten Gutwenger;Michael Jünger;Petra Mutzel;Michael Schulz
Affiliations:
Department of Computer Science, University of Arizona, USA;Department of Computer Science, Technische Universität Dortmund, Germany;Department of Computer Science, University of Cologne, Germany;Department of Computer Science, Technische Universität Dortmund, Germany;Department of Computer Science, University of Cologne, Germany
Venue:
Graph Drawing
Year:
2009

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Abstract

We present a linear-time algorithm for solving the simultaneous embedding problem with fixed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remaining vertices of G, does there exist a planar embedding in which the induced subgraph on each vertex partite of G ∖ C is contained entirely inside or outside C? For the latter problem, we present an algorithm that is based on SPQR-trees and has linear running time. We also show how we can employ SPQR-trees to decide SEFE for two planar graphs where one graph has at most two cycles and the intersection is a pseudoforest in linear time. These results give rise to our hope that our SPQR-tree approach might eventually lead to a polynomial-time algorithm for deciding the general SEFE problem for two planar graphs.