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.
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Gassner, E., Jünger, M., Percan, M., Schaefer, M., Schulz, M. (2006). Simultaneous Graph Embeddings with Fixed Edges. In: Fomin, F.V. (eds) Graph-Theoretic Concepts in Computer Science. WG 2006. Lecture Notes in Computer Science, vol 4271. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11917496_29
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DOI: https://doi.org/10.1007/11917496_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-48381-6
Online ISBN: 978-3-540-48382-3
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