Structural Parameterizations for Equitable Coloring
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An n-vertex graph is equitably k-colorable if there is a proper coloring of its vertices such that each color is used either n/k or n/k times. While classic Vertex Coloring is fixed parameter tractable under well established parameters such as pathwidth and feedback vertex set, equitable coloring is W[1]-hard. We prove that Equitable Coloring is fixed parameter tractable when parameterized by distance to cluster or co-cluster graphs, improving on the FPT algorithm of Fiala et al. (2011) parameterized by vertex cover. In terms of intractability, we adapt the proof of Fellows et al. (2011) to show that Equitable Coloring is W[1]-hard when simultaneously parameterized by distance to disjoint paths and number of colors. We also revisit the literature and derive other results on the parameterized complexity of the problem through minor reductions or other simple observations.
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