Abstract
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:
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Finding a minimum c-connected s–t separator is FPT for c = 2 and W[1]-hard for any c ≥ 3.
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Finding a minimum s–t separator with diameter at most d is W[1]-hard for any d ≥ 2.
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Finding a minimum r-regular s–t separator is W[1]-hard for any r ≥ 1.
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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.
This work is supported by the Research Council of Norway and by the European Research Council (ERC) grant 280152.
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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: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2012. Lecture Notes in Computer Science, vol 7551. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34611-8_33
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