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 analyse how the v-number of \(J_G\) behaves under Gröbner degeneration when G is a closed graph.

分级理想的 v 数是由 Cooper 等人引入的代数不变量，最初是受代数编码理论问题的启发。在本文中，我们研究了二项式边理想的情况，并在它们的 v 数和图中的连通支配概念之间建立了显着的联系。更具体地说，我们证明 v 数位于图G的二项式边理想\(J_G\)的最小素数之一处与定义图的连通支配数一致，提供连通支配​​数的第一代数描述。作为直接推论，我们获得了图的二项式边理想的 v 数的尖锐组合上限。最后，基于边缘理想的一些已知结果，我们分析了当G是闭合图时，\(J_G\)的 v 数在 Gröbner 退化下的表现。