On the Parameterised Complexity of Induced Multipartite Graph Parameters
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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≥2 and ℓ, we must decide whether the maximum value of p over all induced k-partite subgraphs of G is at most ℓ. 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)≤ℓ. We show that for certain parameters this problem is para-NP-hard, while for others it is fixed-parameter tractable.
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