Dominating Ideals and Closed Neighborhood Ideals of Graphs

We study the closed neighborhood ideals and the dominating ideals of graphs, in particular, some classes of trees and cycles. We prove that the closed neighborhood ideals and the dominating ideals of some classes 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) persistance property and the dominating ideals of cycles are nearly normally torsion free.