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 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.