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We introduce a family of graph parameters, called induced multipartite graphparameters, and study their computational complexity. First, we consider thefollowing decision problem: an instance is an induced multipartite graphparameter $p$ 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 isW[1]-hard. Next, we consider a variant of this problem, where we must decidewhether the given graph $G$ contains a sufficiently large induced $k$-partitesubgraph $H$ such that $p(H)\leq\ell$. We show that for certain parameters thisproblem is para-NP-hard, while for others it is fixed-parameter tractable.

On the Parameterised Complexity of Induced Multipartite Graph Parameters