Under ETH, isomorphism-invariant problems cannot be NP-complete on power graphs; Graph Motif is hard on power graphs of cyclic groups, and recognition is polynomial for abelian and some nilpotent power graphs.
Graphs defined on groups
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On the Complexity of Problems on Graphs Defined on Groups
Under ETH, isomorphism-invariant problems cannot be NP-complete on power graphs; Graph Motif is hard on power graphs of cyclic groups, and recognition is polynomial for abelian and some nilpotent power graphs.