On the Parameterised Complexity of Induced Multipartite Graph Parameters

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.