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

Connected domination in graphs and v-numbers of binomial edge ideals