On the parameterized complexity of finding separators with non-hereditary properties

We study the problem of finding small s-t separators that induce graphs having certain properties. It is known that finding a minimum clique s-t separator is polynomial-time solvable (Tarjan 1985), while for example the problems of finding a minimum s-t separator that is a connected graph or an independent set are fixed-parameter tractable (Marx, O'Sullivan and Razgon, manuscript). We extend these results the following way: Finding a minimum c-connected s-t separator is FPT for c∈=∈2 and W[1]-hard for any c∈≥∈3. Finding a minimum s-t separator with diameter at most d is W[1]-hard for any d∈≥∈2. Finding a minimum r-regular s-t separator is W[1]-hard for any r∈≥∈1. For any decidable graph property, finding a minimum s-t separator with this property is FPT parameterized jointly by the size of the separator and the maximum degree. We also show that finding a connected s-t separator of minimum size does not have a polynomial kernel, even when restricted to graphs of maximum degree at most 3, unless NP ⊆ coNP/poly. © 2012 Springer-Verlag Berlin Heidelberg