Simultaneous graph embeddings with fixed edges

Authors:
Elisabeth Gassner;Michael Jünger;Merijam Percan;Marcus Schaefer;Michael Schulz
Affiliations:
Institut für Mathematik B, Technische Universität Graz, Graz, Austria;Institut für Informatik, Universität zu Köln, Köln, Germany;Institut für Informatik, Universität zu Köln, Köln, Germany;School of CTI, DePaul University, Chicago, IL;Institut für Informatik, Universität zu Köln, Köln, Germany
Venue:
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
Year:
2006

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Abstract

We study the problem of simultaneously embedding several graphs on the same vertex set in such a way that edges common to two or more graphs are represented by the same curve. This problem is known as simultaneously embedding graphs with fixed edges. We show that this problem is closely related to the weak realizability problem: Can a graph be drawn such that all edge crossings occur in a given set of edge pairs? By exploiting this relationship we can explain why the simultaneous embedding problem is challenging, both from a computational and a combinatorial point of view. More precisely, we prove that simultaneously embedding graphs with fixed edges is NP-complete even for three planar graphs. For two planar graphs the complexity status is still open.