A set of nodes in the plane is called n-independent if for arbitrary data at those nodes, there is a (not necessarily unique) polynomial of degree at most n that matches the given information. We proved in a previous paper (Hakopian-Toroyan, 2015) that the minimal number of n-independent nodes determining uniquely the curve of degree k≦ n passing through them equals to D:=(1/2)(k-1)(2n+4-k)+2. In this paper we bring a characterization of the case when at least two curves of degree k pass through the nodes of an n-independent node set of cardinality D-1. Namely, we prove that the latter set has a very special construction: All its nodes but one belong to a (maximal) curve of degree k-1. We show that this result readily yields the above cited one. At the end, an important application to the Gasca-Maeztu conjecture is presented.