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