Reducing graph transversals via edge contractions

For a graph invariant π, the Contraction(π) problem consists of, given G and positive integers k,d, deciding whether one can contract k edges of to obtain in which π has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] studied case where is size minimum dominating set. We focus on invariants defined as vertex set that hits all occurrences graphs collection H according fixed containment relation. prove co-NP-hardness results under some assumptions H, particular implying co-NP-hard for k=d=1 when feedback or an odd cycle transversal. In sharp contrast, cover, XP parameterized