A P5-free minimally non-perfectly divisible graph cannot contain a clique cutset, and recognizing perfectly divisible graphs is NP-hard.
Berge, F¨ arbung von Graphen, deren s¨ amtliche bzw
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On the structure of perfectly divisible graphs
A P5-free minimally non-perfectly divisible graph cannot contain a clique cutset, and recognizing perfectly divisible graphs is NP-hard.