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

Finding a Maximum Minimal Separator: Graph Classes and Fixed-Parameter Tractability