Upward Pointset Embeddings of Planar st-Graphs

We study upward pointset embeddings (UPSEs) of planar st-graphs. Let G be a planar st-graph and let \(S \subset \mathbb {R}^2\) 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, if G is an n-vertex planar st-graph whose maximum st-cutset has size k, then UPSE Testing can be solved in \(\mathcal {O}(n^{4k})\) time with \(\mathcal {O}(n^{3k})\) space; also, all the UPSEs of G on S can be enumerated with \(\mathcal {O}(n)\) worst-case delay, using \(\mathcal {O}(k n^{4k} \log n)\) space, after \(\mathcal {O}(k n^{4k} \log n)\) 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 pointset, which can be tested in \(\mathcal {O}(n \log n)\) 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 \(\mathcal {O}(n)\) worst-case delay, using \(\mathcal {O}(n^2)\) space, after \(\mathcal {O}(n^2)\) set-up time.