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.
CITATION STYLE
Heggernes, P., Van’T Hof, P., Marx, D., Misra, N., & Villanger, Y. (2012). On the parameterized complexity of finding separators with non-hereditary properties. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7551 LNCS, pp. 332–343). https://doi.org/10.1007/978-3-642-34611-8_33
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