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Finding a Maximum Minimal Separator: Graph Classes and Fixed-Parameter Tractability
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 that runs in time . 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.Comment: 15 page