We study the problem of finding a maximum cardinality minimal separator of a graph. This problem is known to be NP-hard even for bipartite graphs. In this paper, we strengthen this hardness by showing that for planar bipartite graphs, the problem remains NP-hard. Moreover, for co-bipartite graphs and for line graphs, the problem also remains NP-hard. On the positive side, we give an algorithm deciding whether an input graph has a minimal separator of size at least $k$ that runs in time $2^{O(k)}n^{O(1)}$. We further show that a subexponential parameterized algorithm does not exist unless the Exponential Time Hypothesis (ETH) fails. Finally, we discuss a lower bound for polynomial kernelizations of this problem.

中文翻译：

---

寻找最大最小分隔符：图类和固定参数的可追踪性

我们研究了寻找图的最大基数最小分隔符的问题。即使对于二部图，这个问题也是已知的 NP-hard。在本文中，我们通过表明对于平面二部图，问题仍然是 NP-hard 来加强这种硬度。此外，对于协二部图和线图，问题也仍然是 NP-hard。从积极的方面来说，我们给出了一个算法来决定输入图是否具有大小至少为 $k$ 的最小分隔符，该分隔符在 $2^{O(k)}n^{O(1)}$ 中运行。我们进一步表明，除非指数时间假设 (ETH) 失败，否则不存在次指数参数化算法。最后，我们讨论了这个问题的多项式核化的下界。