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.