We study the problem of finding small s–t separators that induce graphs having certain properties. It is known that finding a minimum clique s–t separator is polynomial‐time solvable (Tarjan in Discrete Math. 55:221–232, 1985), while for example the problems of finding a minimum s–t separator that induces a connected graph or forms an independent set are fixed‐parameter tractable when parameterized by the size of the separator (Marx et al. in ACM Trans. Algorithms 9(4): 30, 2013). Motivated by these results, we study properties that generalize cliques, independent sets, and connected graphs, and determine the complexity of finding separators satisfying these properties. We investigate these problems also on bounded‐degree graphs. Our results are as follows:
Finding a minimum c‐connected s–t separator is FPT for c=2 and W[1]‐hard for any c≥3.
Finding a minimum s–t separator with diameter at most d is W[1]‐hard for any d≥2.
Finding a minimum r‐regular s–t separator is W[1]‐hard for any r≥1.
For any decidable graph property, finding a minimum s–t separator with this property is FPT parameterized jointly by the size of the separator and the maximum degree.
Finding a connected s–t separator of minimum size does not have a polynomial kernel, even when restricted to graphs of maximum degree at most 3, unless \(\mathrm{NP} \subseteq \mathrm{coNP/poly}\).
In order to prove (1), we show that the natural c‐connected generalization of the well‐known Steiner Tree problem is FPT for c=2 and W[1]‐hard for any c≥3.
