Upward Book Embeddability of st-Graphs: Complexity and Algorithms

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