Star Decomposition is para-NP-hard for several structural parameters, W[1]-hard for vertex cover number, FPT for (vc,d), (td,s), and (nd,s), and XP for treewidth and vc.
On star partition of split graphs
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Parameterized Complexity of the Star Decomposition Problem
Star Decomposition is para-NP-hard for several structural parameters, W[1]-hard for vertex cover number, FPT for (vc,d), (td,s), and (nd,s), and XP for treewidth and vc.