Mixed graph coloring is W[1]-hard parameterized by treewidth and paraNP-hard by neighborhood diversity, but FPT parameterized by the introduced mixed neighborhood diversity.
On the clique-width of some perfect graph classes
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The Parameterized Complexity of Coloring Mixed Graphs
Mixed graph coloring is W[1]-hard parameterized by treewidth and paraNP-hard by neighborhood diversity, but FPT parameterized by the introduced mixed neighborhood diversity.