Open Neighborhood Ideals of Well Totally Dominated Trees are Cohen-Macaulay

2025/10/22 by Lim, Jounglag, Gossell, James, Sather-Wagstaff, Keri Ann +8 · 1 citation
#05C69 #05E40 #13H10 #Commutative Algebra (math.AC) #FOS: Mathematics #Primary: 13F55 #Secondary: 05C05

paper · doi:10.48550/arxiv.2510.19707

Abstract

We introduce and investigate the open neighborhood ideal N(G) of a finite simple graph G. We describe the minimal primary decomposition of N(G) in terms of the minimal total dominating sets (TD-sets) of G. Then we prove that the open neighborhood ideal of a tree is Cohen-Macaulay if and only if the tree is unmixed (well totally dominated) and calculate the Cohen-Macaulay type. We also give a descriptive characterization of all unmixed trees which takes polynomial time to verify.

Citations

• Closed Neighborhood Ideals of Finite Simple Graphs
• On well-dominated graphs
• Depths and Cohen-Macaulay Properties of Path Ideals
• Shellable graphs and sequentially Cohen-Macaulay bipartite graphs
• Cohen-Macaulay chordal graphs

Cited by

• Geometrically vertex decomposable open neighborhood ideals

Related

• Closed Neighborhood Ideals of Finite Simple Graphs
• Geometrically vertex decomposable open neighborhood ideals
• Depths and Cohen-Macaulay Properties of Path Ideals
• Barile-Macchia Resolutions and the closed neighborhood ideal
• Dominating ideals and closed neighborhood ideals of graphs