小簇的NodeTrix平面性测试
NodeTrix Planarity Testing with Small Clusters
摘要：
本文研究了当每个簇的最大尺寸由一个常数$k$限定时，平坦簇图的NodeTrix平面性检验问题。我们认为这两种情况时，双方的矩阵的边缘是入射固定的情况下，当他们可以任意选择。我们发现，NodeTrix平面性测试与固定的双方可以解决在$O(k^{3k+\frac{3}{2}} \cdot n)$时间为每个平面聚类图，可以减少到一个部分2-树，通过折叠成单顶点的集群。在一般情况下，对于$k = 2$，固定边的NodeTrix平面性测试可以在$O(n)$时间内解决，但对于任何$k > 2$，它都是NP完全的。当使用$k > 4$时，NodeTrix平面性测试在自由侧模型中也保持NP完全。
英文摘要：
We study the NodeTrix planarity testing problem for flat clustered graphs when the maximum size of each cluster is bounded by a constant $k$. We consider both the case when the sides of the matrices to which the edges are incident are fixed and the case when they can be chosen arbitrarily. We show that NodeTrix planarity testing with fixed sides can be solved in $O(k^{3k+\frac{3}{2}} \cdot n)$ time for every flat clustered graph that can be reduced to a partial 2-tree by collapsing its clusters into single vertices. In the general case, NodeTrix planarity testing with fixed sides can be solved in $O(n)$ time for $k = 2$, but it is NP-complete for any $k > 2$. NodeTrix planarity testing remains NP-complete also in the free sides model when $k > 4$.
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