Final answer:
The centrioles do NOT have a role in compacting DNA, unlike nucleosomes and chromatin.
Step-by-step explanation:
The structure that does NOT have a role in compacting DNA is centrioles. Centrioles are involved in cell division and play a role in the formation of the mitotic spindle, but they are not directly involved in compacting DNA. On the other hand, both nucleosomes and chromatin play important roles in compacting DNA. Nucleosomes are made up of DNA wrapped around histone proteins, and they help to package the DNA into a more condensed form. Chromatin, which is a complex of DNA, histones, and other proteins, further helps to compact DNA into the tightly-packed structure seen in chromosomes.
Learn more about role of different structures in compacting DNA