LIPIcs.GD.2024.24.pdf
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We study upward pointset embeddings (UPSEs) of planar st-graphs. Let G be a planar st-graph and let S ⊂ ℝ² be a pointset with |S| = |V(G)|. An UPSE of G on S is an upward planar straight-line drawing of G that maps the vertices of G to the points of S. We consider both the problem of testing the existence of an UPSE of G on S (UPSE Testing) and the problem of enumerating all UPSEs of G on S. We prove that UPSE Testing is NP-complete even for st-graphs that consist of a set of directed st-paths sharing only s and t. On the other hand, for n-vertex planar st-graphs whose maximum st-cutset has size k, we prove that it is possible to solve UPSE Testing in 𝒪(n^{4k}) time with 𝒪(n^{3k}) space, and to enumerate all UPSEs of G on S with 𝒪(n) worst-case delay, using 𝒪(k n^{4k} log n) space, after 𝒪(k n^{4k} log n) set-up time. Moreover, for an n-vertex st-graph whose underlying graph is a cycle, we provide a necessary and sufficient condition for the existence of an UPSE on a given poinset, which can be tested in 𝒪(n log n) time. Related to this result, we give an algorithm that, for a set S of n points, enumerates all the non-crossing monotone Hamiltonian cycles on S with 𝒪(n) worst-case delay, using 𝒪(n²) space, after 𝒪(n²) set-up time.
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