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.

翻译：我们引入了一族图参数，称为诱导多部图参数，并研究了其计算复杂性。首先，考虑以下判定问题：实例包括一个诱导多部图参数$p$和给定图$G$，对于自然数$k\geq2$和$\ell$，需要判定$G$的所有诱导$k$-部子图上的最大值$p$是否不超过$\ell$。我们证明该问题是W[1]-难的。其次，考虑该问题的一个变种，其中需要判定给定图$G$是否包含一个足够大的诱导$k$-部子图$H$，使得$p(H)\leq\ell$。我们表明，对于某些参数，该问题是para-NP-难的，而对于其他参数，它是固定参数易处理的。