An SPQR-Tree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges
Fowler, J. Joseph and Gutwenger, Carsten and Jünger, Michael and Mutzel, Petra and Schulz, Michael (2009) An SPQR-Tree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges. In: Graph Drawing 16th International Symposium, GD 2008, September 21- 24, 2008, Heraklion, Crete, Greece , pp. 157-168 (Official URL: http://dx.doi.org/10.1007/978-3-642-00219-9_16).
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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.
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