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We study the closed neighborhood ideals and the dominating ideals of graphs,in particular, of trees and cycles. We prove that the closed neighborhoodideals and the dominating ideals of trees are normally torsion-free. The closedneighborhood ideals and the dominating ideals of cycles fail to be normallytorsion-free. However, we prove that the closed neighborhood ideals of cyclesadmit the (strong) persistence property and the dominating ideals of cycles arenearly normally torsion-free.

Dominating ideals and closed neighborhood ideals of graphs