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

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.

• Type: Conference
• Year: 2008
• Where: GD
• Authors: J. Joseph Fowler, Carsten Gutwenger, Michael Jünger, Petra Mutzel, Michael Schulz
• Keywords: Computer Graphics, Embedding Problem, GD 2008, Planar Graph G, Planar Graphs