Closed Neighborhood Ideals of Finite Simple Graphs

TL;DR

This paper studies the algebraic properties of closed neighborhood ideals in finite simple graphs, providing explicit decompositions and characterizations of Cohen-Macaulay trees, advancing understanding in combinatorial commutative algebra.

Contribution

It explicitly describes minimal irreducible decompositions of closed neighborhood ideals and characterizes Cohen-Macaulay trees, independent of the field's characteristic.

Findings

Explicit minimal irreducible decompositions of the ideals

Characterization of Cohen-Macaulay trees

Property is characteristic independent for trees

Abstract

We investigate Sharifan and Moradi's closed neighborhood ideal of a finite simple graph, which is a square-free monomial ideal in a polynomial ring over a field. We explicitly describe the minimal irreducible decompositions of these ideals. We also characterize the trees whose closed neighborhood ideals are Cohen-Macaulay; in particular, this property for closed neighborhood ideals of trees is characteristic independent.