  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.
