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.

翻译：本文研究分层平面图绘制中聚类平面性问题的变体。迄今为止，该领域仅研究了凸聚类问题。我们引入两种新变体，两者均要求输入图的分层平面绘制，但放松了对聚类形状的约束。在无约束聚类分层平面性（uCLP）问题中，我们仅要求聚类由简单闭合曲线包围，该曲线恰好包含聚类顶点，且与图的每条边至多相交一次。在单调聚类分层平面性（y-CLP）问题中，额外要求每个聚类能够通过不穿越聚类边界的边进行增强，使得该聚类在保持图分层平面性的同时连通，从而在分层平面场景中模拟经典聚类平面性的特征。对于双连通且具有单一源点的输入图，我们给出了uCLP问题的多项式时间算法。相比之下，我们证明在相同约束下y-CLP问题是困难的，且即使当层数被常数界约束且仅存在单个非平凡聚类时，该问题仍保持NP难性。