TY - JOUR

T1 - Parameterizing cut sets in a graph by the number of their components

AU - Ito, Takehiro

AU - Kamiński, Marcin

AU - Paulusma, Daniël

AU - Thilikos, Dimitrios M.

N1 - Funding Information:
The first author was supported by Grant-in-Aid for Young Scientists (B) 22700001. The second author was supported by Chargé de Recherches du F.R.S. – FNRS. The third author was supported by EPSRC (EP/D053633/1) and the last author was supported by the project ‘‘Kapodistrias’’ (AΠ 02839/28.07.2008) of the National and Kapodistrian University of Athens (project code: 70/4/8757).

PY - 2011/10/21

Y1 - 2011/10/21

N2 - 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, ℓ)-cut if V \ U induces a subgraph with exactly ℓ(≥ 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, ℓ) -CUT are to test whether a graph has a k-cut or (k, ℓ) -cut, respectively. By pinpointing a close relationship to graph contractibility problems we show that (k, ℓ) -CUT is in P for k = 1 and any fixed constant ℓ ≥ 2, while it is NP-complete for any fixed pair k, ℓ ≥ 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, ℓ)-CUT on graphs of Euler genus at most g when parameterized by k +ℓ. 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.

AB - 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, ℓ)-cut if V \ U induces a subgraph with exactly ℓ(≥ 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, ℓ) -CUT are to test whether a graph has a k-cut or (k, ℓ) -cut, respectively. By pinpointing a close relationship to graph contractibility problems we show that (k, ℓ) -CUT is in P for k = 1 and any fixed constant ℓ ≥ 2, while it is NP-complete for any fixed pair k, ℓ ≥ 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, ℓ)-CUT on graphs of Euler genus at most g when parameterized by k +ℓ. 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.

KW - 2K2-partition

KW - Cut set

KW - Graph contractibility

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U2 - 10.1016/j.tcs.2011.07.005

DO - 10.1016/j.tcs.2011.07.005

M3 - Article

AN - SCOPUS:84864123827

VL - 412

SP - 6340

EP - 6350

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 45

ER -