Private neighbors, perfect codes and their relation with the $\mathtt{v}$-number of closed neighborhood ideals

In this work, we investigate the connections between dominating sets, private neighbors, and perfect codes in graphs, and their relationships with commutative algebra. In particular, we estimate the $\mathtt{v}$-number of closed neighborhood ideals in terms of minimal dominating sets and private neighbors. We show how the $\mathtt{v}$-number is related to other graph invariants, such as the cover number, domination number, and matching number. Moreover, we explore the relation with the Castelnuovo-Mumford regularity, proving that the $\mathtt{v}$-number is a lower bound for the regularity of bipartite, very well-covered, and chordal graphs. Finally, drawing from the relation between efficient dominating set and perfect codes, we use the redundancy of Hamming codes to present lower and upper bounds for the $\mathtt{v}$-number of some special family of graphs.