Upward Partitioned Book Embeddings
We analyze a directed variation of the book embedding problem when the page partition is prespecified and the nodes on the spine must be in topological order (upward book embedding). Given a directed acyclic graph and a partition of its edges into k pages, can we linearly order the vertices such that the drawing is upward (a topological sort) and each page avoids crossings? We prove that the problem is NP-complete for \(k\ge 3\), and for \(k\ge 4\) even in the special case when each page is a matching. By contrast, the problem can be solved in linear time for \(k=2\) pages when pages are restricted to matchings. The problem comes from Jack Edmonds (1997), motivated as a generalization of the map folding problem from computational origami.
We thank Jack Edmonds for valuable discussions in August 1997 where he described how Upward Matching-Partitioned k -Page Book Embedding generalizes the map folding problem. We also thank Therese Biedl for valuable discussions in 2007 about the complexity this problem.
This research was conducted during the 31st Bellairs Winter Workshop on Computational Geometry which took place in Holetown, Barbados on March 18–25, 2016. We thank the other participants of the workshop for helpful discussion and for providing a fun and stimulating environment. We also thank our anonymous referees for helpful suggestions in improving the clarity of our paper.
Supported in part by the NSF award CCF-1422311 and Science without Borders. Quanquan Liu is supported in part by NSF GRFP under Grant No. (1122374).
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