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.