Geometrically vertex decomposable open neighborhood ideals

Abstract

In this paper, we prove that the Stanley-Reisner complexes of open neighborhood ideals of TD-unmixed trees are vertex decomposable. In particular, we show that these open neighborhood ideals are geometric vertex decomposable.
We further demonstrate that Cohen-Macaulay open neighborhood ideals of trees are special cases of Cohen-Macaulay facet ideals of simplicial trees. Finally, we investigate open neighborhood ideals of chordal graphs and establish that almost all square-free monomial ideals can be realized as the open neighborhood ideal of a chordal graph.