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

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 testwhether a graph has a disconnected cut and is known to beNP-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 thatDisconnected Cut is solvable in polynomial time for minor-closed classes of graphs excluding some apex graph. © 2011 Elsevier B.V. All rights reserved.