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

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