Let G = (V, E) be a directed graph and : V → [k] := {1, . . . , k} a level assignment such that (u) < (v) for all directed edges (u, v) ∈ E. A level planar drawing of G is a drawing of G where each vertex v is mapped to a unique point on the horizontal line i with y-coordinate (v), and each edge is drawn as a y-monotone curve between its endpoints such that no two curves cross in their interior.In the problem Constrained Level Planarity (CLP for short), we are further given a partial order-and we seek a level planar drawing where the order of the vertices on i is a linear extension of ≺ i . A special case of this is the problem Partial Level Planarity (PLP for short), where we are asked to extend a given level-planar drawing H of a subgraph H ⊆ G to a complete drawing G of G without modifying the given drawing, i.e., the restriction of G to H must coincide with H.We give a simple polynomial-time algorithm with running time O(n 5 ) for CLP of single-source graphs that is based on a simplified version of an existing levelplanarity testing algorithm for single-source graphs. We introduce a modified type of PQ-tree data structure that is capable of efficiently handling the arising constraints to improve the running time to O(n + k ), where is the size of the constraints. We complement this result by showing that PLP is NP-complete even in very restricted cases. In particular, PLP remains NPcomplete even when G is a subdivision of a triconnected planar graph with bounded degree.

We show that finding orthogonal grid-embeddings of plane graphs (planar with fixed combinatorial embedding) with the minimum number of bends in the so-called Kandinsky model (which allows vertices of degree > 4) is NP-complete, thus solving a long-standing open problem. On the positive side, we give an efficient algorithm for several restricted variants, such as graphs of bounded branch width and a subexponential exact algorithm for general plane graphs.

We introduce and study level-planar straight-line drawings with a fixed number λ of slopes. For proper level graphs, we give an O(n log 2 n/ log log n)-time algorithm that either finds such a drawing or determines that no such drawing exists. Moreover, we consider the partial drawing extension problem, where we seek to extend an immutable drawing of a subgraph to a drawing of the whole graph, and the simultaneous drawing problem, which asks about the existence of drawings of two graphs whose restrictions to their shared subgraph coincide. We present O(n 4/3 log n)-time and O(λn 10/3 log n)-time algorithms for these respective problems on proper level-planar graphs. We complement these positive results by showing that testing whether non-proper level graphs admit level-planar drawings with λ slopes is NPhard even in restricted cases.

The SPQR-tree is a data structure that compactly represents all planar embeddings of a biconnected planar graph. It plays a key role in constrained planarity testing. We develop a similar data structure, called the UP-tree, that compactly represents all upward planar embeddings of a biconnected single-source directed graph. We demonstrate the usefulness of the UP-tree by solving the upward planar embedding extension problem for biconnected singlesource directed graphs.

Recently, Fulek et al. [5,6,7] have presented Hanani-Tutte results for (radial) level planarity, i.e., a graph is (radial) level planar if it admits a (radial) level drawing where any two (independent) edges cross an even number of times. We show that the 2-Sat formulation of level planarity testing due to Randerath et al. [14] is equivalent to the strong Hanani-Tutte theorem for level planarity [7]. Further, we show that this relationship carries over to radial level planarity, which yields a novel polynomial-time algorithm for testing radial level planarity.