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

Abstract

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 $c\geq 1$, W[$t$]-hard for all $t\in \mathbb{N}$ 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