Fair matching is FPT for treewidth plus maximum right-degree, neighborhood diversity, and tree-depth plus number of colors, yet W[1]-hard for feedback vertex number plus tree-depth and for path-width plus number of colors.
Matroids, matchings, and fairness
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Parameterized Complexity of Fair Many-to-One Matchings
Fair matching is FPT for treewidth plus maximum right-degree, neighborhood diversity, and tree-depth plus number of colors, yet W[1]-hard for feedback vertex number plus tree-depth and for path-width plus number of colors.