We study k-page upward book embeddings (kUBEs) of st-graphs, that is, book embeddings of single-source single-sink directed acyclic graphs on k pages with the additional requirement that the vertices of the graph appear in a topological ordering along the spine of the book. We show that testing whether a graph admits a kUBE is NP-complete for k ≥ 3. A hardness result for this problem was previously known only for k = 6 [Heath and Pemmaraju, 1999]. Motivated by this negative result, we focus our attention on k = 2. On the algorithmic side, we present polynomial-time algorithms for testing the existence of 2UBEs of planar st-graphs with branchwidth β and of plane st-graphs whose faces have a special structure. These algorithms run in O(f(β) · n + n3) time and O(n) time, respectively, where f is a singly-exponential function on β. Moreover, on the combinatorial side, we present two notable families of plane st-graphs that always admit an embedding-preserving 2UBE.
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