Parameterizing cut sets in a graph by the number of their components

  • Authors:

  • Takehiro Ito;Marcin Kamiski;Daniël Paulusma;Dimitrios M. Thilikos

  • Affiliations:

  • Graduate School of Information Sciences, Tohoku University, Aoba-yama 6-6-05, Sendai, 980-8579, Japan;Computer Science Department, Université Libre de Bruxelles, Boulevard du Triomphe CP212, B-1050 Brussels, Belgium;Department of Computer Science, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, England, United Kingdom;Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, GR15784 Athens, Greece

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2011

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Abstract

For a connected graph G=(V,E), a subset U@?V is a disconnected cut if U disconnects G and the subgraph G[U] induced by U is disconnected as well. A cut U is a k-cut if G[U] contains exactly k(=1) components. More specifically, a k-cut U is a (k,@?)-cut if V@?U induces a subgraph with exactly @?(=2) components. The Disconnected Cut problem is to test whether a graph has a disconnected cut and is known to be NP-complete. The problems k-Cut and (k,@?)-Cut are to test whether a graph has a k-cut or (k,@?)-cut, respectively. By pinpointing a close relationship to graph contractibility problems we show that (k,@?)-Cut is in P for k=1 and any fixed constant @?=2, while it is NP-complete for any fixed pair k,@?=2. We then prove that k-Cut is in P for k=1 and NP-complete for any fixed k=2. On the other hand, for every fixed integer g=0, we present an FPT algorithm that solves (k,@?)-Cut on graphs of Euler genus at most g when parameterized by k+@?. By modifying this algorithm we can also show that k-Cut is in FPT for this graph class when parameterized by k. Finally, we show that Disconnected Cut is solvable in polynomial time for minor-closed classes of graphs excluding some apex graph.