Upward Partitioned Book Embeddings

TL;DR

This paper investigates the computational complexity of upward book embeddings with a fixed page partition, proving NP-completeness for three or more pages and providing efficient solutions for two pages.

Contribution

It establishes the NP-completeness of upward book embedding with three or more pages and offers a linear-time algorithm for two pages with matchings.

Findings

NP-complete for k ≥ 3 pages

NP-complete even when pages are matchings for k ≥ 4

Linear-time solution for k=2 pages with matchings

Abstract

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.