Upward Book Embeddability of st -Graphs: Complexity and Algorithms

k - page upward book embedding ( k UBE) 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 k UBE Testing problem, which asks whether a graph admits a k UBE, 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 and NP-complete for 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 k UBE Testing is NP-complete for 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 k UBE 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(β )· n+n^3) -time algorithm for 2UBE Testing , where β is the branchwidth of the input graph and f is a singly-exponential function on 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 UBE.