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

  • Authors:

  • Pinar Heggernes;Pim van't Hof;Dániel Marx;Neeldhara Misra;Yngve Villanger

  • Affiliations:

  • Department of Informatics, University of Bergen, Norway;Department of Informatics, University of Bergen, Norway;Computer and Automation Research Institute, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary;Institute of Mathematical Sciences, Chennai, India;Department of Informatics, University of Bergen, Norway

  • Venue:
  • WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
  • Year:
  • 2012

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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: · 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.