Clustered Planarity Variants for Level Graphs

We consider variants of the clustered planarity problem for level-planar drawings. So far, only convex clusters have been studied in this setting. We introduce two new variants that both insist on a level-planar drawing of the input graph but relax the requirements on the shape of the clusters. In unrestricted Clustered Level Planarity (uCLP) we only require that they are bounded by simple closed curves that enclose exactly the vertices of the cluster and cross each edge of the graph at most once. The problem y-monotone Clustered Level Planarity (y-CLP) requires that additionally it must be possible to augment each cluster with edges that do not cross the cluster boundaries so that it becomes connected while the graph remains level-planar, thereby mimicking a classic characterization of clustered planarity in the level-planar setting. We give a polynomial-time algorithm for uCLP if the input graph is biconnected and has a single source. By contrast, we show that y-CLP is hard under the same restrictions and it remains NP-hard even if the number of levels is bounded by a constant and there is only a single non-trivial cluster.