Parameterizing Cut Sets in a Graph by the Number of Their Components

  • Authors:

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

  • Affiliations:

  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan 980-8579;Computer Science Department, Université Libre de Bruxelles, Brussels, Belgium B-1050;Department of Computer Science, University of Durham, Science Laboratories, Durham, England DH1 3LE;Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece GR15784

  • Venue:
  • ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
  • Year:
  • 2009

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Abstract

For a connected graph G = (V,E), a subset U 驴 V is called a k-cut if U disconnects G, and the subgraph induced by U contains exactly k ( 驴 1) components. More specifically, a k-cut U is called a (k,驴)-cut if V \U induces a subgraph with exactly 驴 ( 驴 2) components. We study two decision problems, called k-Cut and (k,驴)-Cut, which determine whether a graph G has a k-cut or (k,驴)-cut, respectively. By pinpointing a close relationship to graph contractibility problems we first show that (k,驴)-Cut is in P for k = 1 and any fixed constant 驴 驴 2, while the problem is NP-complete for any fixed pair k,驴 驴 2. We then prove that k-Cut is in P for k = 1, and is NP-complete for any fixed k 驴 2. On the other hand, we present an FPT algorithm that solves (k,驴)-Cut on apex-minor-free graphs when parameterized by k + 驴. By modifying this algorithm we can also show that k-Cut is in FPT (with parameter k) and Disconnected Cut is solvable in polynomial time for apex-minor-free graphs. The latter problem asks if a graph has a k-cut for some k 驴 2.