层级图的聚类平面性变体
Clustered Planarity Variants for Level Graphs
AI总结：
本文研究层级平面绘图中聚类形状要求放宽后的两种聚类平面性变体，给出uCLP在双连通单源图上的多项式算法，并证明y-CLP在同样限制下仍为NP难。
摘要：
我们考虑层级平面绘图中聚类平面性问题的变体。迄今为止，在该设定下仅研究了凸聚类。我们引入两种新变体，二者都要求输入图具有层级平面绘图，但放宽了对聚类形状的要求。在无限制聚类层级平面性（uCLP）中，我们仅要求聚类由简单闭曲线界定，这些曲线恰好包围该聚类的顶点，并且与图的每条边最多相交一次。y-单调聚类层级平面性（y-CLP）问题则额外要求：必须能够为每个聚类增补不穿越聚类边界的边，使其连通，同时图保持层级平面，从而在层级平面设定中模拟经典聚类平面性的一个刻画。我们针对输入图双连通且具有单一源点的情形，给出了uCLP的多项式时间算法。相比之下，我们证明在相同限制下y-CLP是困难的，并且即使层数被常数界定且仅存在一个非平凡聚类，该问题仍保持NP难。
英文摘要：
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.
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