Normality and associated primes of closed neighborhood ideals and dominating ideals

Abstract

In this paper, we first give some sufficient criteria for normality of monomial ideals. As applications, we show that closed neighborhood ideals of complete bipartite graphs are normal, and hence satisfy the (strong) persistence property. We also prove that dominating ideals of complete bipartite graphs are nearly normally torsion-free. In addition, we show that dominating ideals of [Formula: see text]-wheel graphs, under certain condition, are normal.

Classes of normally and nearly normally torsion-free monomial ideals

“…It is shown in [20,Lemma 2.2] that DI(G) is the Alexander dual of NI(G), that is, DI(G) = NI(G) ∨ . In order to find more properties of closed neighborhood ideals and dominating ideals, the reader may refer to [9,14,20].…”

An algebraic approach to sets defining minimal dominating sets of regular graphs

“…After that, in [3], Honeycutt and Sather-Wagstaff probed the Cohen-Macaulay, unmixed, and complete intersection properties of closed neighborhood ideals. Next, in [9,7], the authors concentrated on the normality, strong persistence property, persistence property, and symbolic strong persistence property of closed neighborhood ideals and dominating ideals of some classes of graphs.…”