We study Clustered Planarity with Linear Saturators, which is the problem of augmenting an $n$-vertex planar graph whose vertices are partitioned into independent sets (called clusters) with paths - one for each cluster - that connect all the vertices in each cluster while maintaining planarity. We show that the problem can be solved in time $2^{O(n)}$ for both the variable and fixed embedding case. Moreover, we show that it can be solved in subexponential time $2^{O(\sqrt{n}\log n)}$ in the fixed embedding case if additionally the input graph is connected. The latter time complexity is tight under the Exponential-Time Hypothesis. We also show that $n$ can be replaced with the vertex cover number of the input graph by providing a linear (resp. polynomial) kernel for the variable-embedding (resp. fixed-embedding) case; these results contrast the NP-hardness of the problem on graphs of bounded treewidth (and even on trees). Finally, we complement known lower bounds for the problem by showing that Clustered Planarity with Linear Saturators is NP-hard even when the number of clusters is at most $3$, thus excluding the algorithmic use of the number of clusters as a parameter.

翻译：我们研究具有线性饱和器的聚类平面性问题，该问题旨在对$n$个顶点的平面图进行增广：给定顶点被划分为独立集（称为聚类），需为每个聚类添加一条路径，使得该路径连接该聚类中的所有顶点，同时保持平面性。我们证明该问题在变嵌入和固定嵌入情形下均可在$2^{O(n)}$时间内求解。此外，我们证明在固定嵌入情形下，若输入图连通，则可在亚指数时间$2^{O(\sqrt{n}\log n)}$内求解。基于指数时间假设，后一时间复杂度是紧的。我们还证明可通过为变嵌入（对应固定嵌入）情形提供线性（对应多项式）核，将$n$替换为输入图的顶点覆盖数；这些结果与问题在有界树宽图（甚至在树上）的NP难性形成对比。最后，我们通过证明即使聚类数量至多为$3$时具有线性饱和器的聚类平面性仍是NP难问题，补充了该问题的已知下界，从而排除了将聚类数量作为算法参数的可能性。