The v-number of a graded ideal is an invariant recently introduced in the context of coding theory, particularly in the study of Reed–Muller-type codes. In this work, we study the localized v-numbers of a binomial edge ideal \(J_G\) associated to a finite simple graph G. We introduce a new approach to compute these invariants, based on the analysis of transversals in families of subsets arising from dependencies in certain rank-two matroids. This reduces the computation of localized v-numbers to the determination of the radical of an explicit ideal and provides upper bounds for these invariants. Using this method, we explicitly compute the localized v-numbers of \(J_G\) at the associated minimal primes corresponding to minimal cuts of G. Additionally, we determine the v-number of binomial edge ideals for cycle graphs and give an almost complete answer to a conjecture from Dey et al. (Int J Algebra Comput 35(01):119–143, 2024), showing that the v-number of a cycle graph \(C_n\) is either \(\textstyle \big \lceil \frac{2n}{3} \big \rceil \) or \(\textstyle \big \lceil \frac{2n}{3} \big \rceil - 1\) .
