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