A k-page upward book embedding (kUBE) of a directed acyclic graph G is a book embeddings of G on k pages with the additional requirement that the vertices appear in a topological ordering along the spine of the book. The kUBE Testing problem, which asks whether a graph admits a kUBE, was introduced in 1999 by Heath, Pemmaraju, and Trenk (SIAM J Comput 28(4), 1999). In a companion paper, Heath and Pemmaraju (SIAM J Comput 28(5), 1999) proved that the problem is linear-time solvable for $$k=1$$
k
=
1
and NP-complete for $$k = 6$$
k
=
6
. Closing this gap has been a central question in algorithmic graph theory since then. In this paper, we make a major contribution towards a definitive answer to the above question by showing that kUBE Testing is NP-complete for $$k\ge 3$$
k
≥
3
, even for st-graphs, i.e., acyclic directed graphs with a single source and a single sink. Indeed, our result, together with a recent work of Bekos et al. (Theor Comput Sci 946, 2023) that proves the NP-completeness of 2UBE for planar st-graphs, closes the question about the complexity of the kUBE problem for any k. Motivated by this hardness result, we then focus on the 2UBE Testing for planar st-graphs. On the algorithmic side, we present an $$O(f(\beta )\cdot n+n^3)$$
O
(
f
(
β
)
·
n
+
n
3
)
-time algorithm for 2UBE Testing, where $$\beta $$
β
is the branchwidth of the input graph and f is a singly-exponential function on $$\beta $$
β
. Since the treewidth and the branchwidth of a graph are within a constant factor from each other, this result immediately yields an FPT algorithm for st-graphs of bounded treewidth. Furthermore, we describe an O(n)-time algorithm to test whether a plane st-graph whose faces have a special structure admits a 2UBE that additionally preserves the plane embedding of the input st-graph. On the combinatorial side, we present two notable families of plane st-graphs that always admit an embedding-preserving $$2$$
2
UBE.
