The Parameterized Complexity Binary CSP for Graphs with a Small Vertex Cover and Related Results

08/26/2022

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by   Hans L. Bodlaender, et al.

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In this paper, we show that Binary CSP with the size of a vertex cover as parameter is complete for the class W[3]. We obtain a number of related results with variations of the proof techniques, that include: Binary CSP is complete for W[2d+1] with as parameter the size of a vertex modulator to graphs of treedepth c, or forests of depth d, for constant c≥ 1, W[t]-hard for all t∈ℕ with treewidth as parameter, and hard for W[SAT] with feedback vertex set as parameter. As corollaries, we give some hardness and membership problems for classes in the W-hierarchy for List Colouring under different parameterisations.