Decompositions of functions defined on finite sets in ${\mathbb{R}}^d$

A finite subset $M\subset {\mathbb{R}}^d$ is basic, if for any function $f \colon M \to $\mathbb{R}$$thereexistsacollectionoffunctions$f_1, \ldots, f_d \colon $\mathbb{R}$ \to $\mathbb{R}$$suchthatforeachelement$(x_1, \ldots, x_d)\in M$wehave$f(x_1, \ldots, x_d) = f_1(x_1) + \ldots + f_d(x_d)$. For certain finite sets, we prove a criterion for a set to be basic, and we show that it cannot be extended to the general case. In addition, we interpret the above criterion in terms of doubly-weighted graphs and give an estimation for the number of elements in certain basic and non-basic subsets.