  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 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 $d=1$. When the operation is edge
deletion and the parameter is the chromatic number, we determine the
computational complexity of the associated problem on cographs and complete
multipartite graphs. Our results answer several open questions stated in [Diner
et al., Theoretical Computer Science, 746, p. 49-72 (2012)].
