  We introduce a family of graph parameters, called induced multipartite graph
parameters, and study their computational complexity. First, we consider the
following decision problem: an instance is an induced multipartite graph
parameter $p$ and a given graph $G$, and for natural numbers $k\geq2$ and
$\ell$, we must decide whether the maximum value of $p$ over all induced
$k$-partite subgraphs of $G$ is at most $\ell$. We prove that this problem is
W[1]-hard. Next, we consider a variant of this problem, where we must decide
whether the given graph $G$ contains a sufficiently large induced $k$-partite
subgraph $H$ such that $p(H)\leq\ell$. We show that for certain parameters this
problem is para-NP-hard, while for others it is fixed-parameter tractable.
