  We consider the following problem: for a given graph G and two integers k and
d, can we apply a fixed graph operation at most k times in order to reduce a
given graph parameter $\pi$ by at least d? We show that this problem is NP-hard
when the parameter is the independence number and the graph operation is vertex
deletion or edge contraction, even for fixed d=1 and when restricted to chordal
graphs. We also give a polynomial time algorithm for bipartite graphs when the
operation is edge contraction, the parameter is the independence number and d
is fixed. Further, we complete the complexity dichotomy on H-free graphs when
the parameter is the clique number and the operation is edge contraction by
showing that this problem is NP-hard in ($C_3+P_1$)-free graphs even for fixed
d=1. Our results answer several open questions stated in [Diner et al.,
Theoretical Computer Science, 746, p. 49-72 (2012)].
