  The v-number of a graded ideal is an algebraic invariant introduced by Cooper
et al., and originally motivated by problems in algebraic coding theory. In
this paper we study the case of binomial edge ideals and we establish a
significant connection between their v-numbers and the concept of connected
domination in graphs. More specifically, we prove that the localization of the
v-number at one of the minimal primes of the binomial edge ideal $J_G$ of a
graph $G$ coincides with the connected domination number of the defining graph,
providing a first algebraic description of the connected domination number. As
an immediate corollary, we obtain a sharp combinatorial upper bound for the
v-number of binomial edge ideals of graphs. Lastly, building on some known
results on edge ideals, we analyze how the v-number of $J_G$ behaves under
Gr\"obner degeneration when $G$ is a closed graph.
