Given a set of independence relations, how can I find a set of vectors satisfying them? Let $G = (V,E)$ be a graph. The problem is to find a set of vectors $W$ such that there exists some bijection $f:V \to W$ and for all $a \in V$, we have $f(a)$ is not a linear combination of $f(N(a))$, the image of its neighbors. 
For a specific example, consider the graph $K_{2,2,2}$, the 1-skeleton of the octahedron. I can obviously compute a set by hand, but this seems like something that has been done before so I'm looking for references. I'm also interested in a projective version of this question. 
Thanks for your help and let me know if I should clarify anything.
 A: A natural additional condition is that for each vertex $v$ the set $\{v\} \cup N(v)$ forms a maximal independent set. Let $\Delta = \Delta(G)$ be the simplicial complex whose facets are the sets $\{v\} \cup N(v)$.
Note that in general $\Delta(G)$ is neither the independence complex of $G$ nor the matroid independence complex of $G$. 
Now, our new simplicial complex $\Delta$ is representable by a collection of vectors in some vector space if and only if $\Delta$ is the matroid independence complex of some realizable matroid. In particular, all facets of $\Delta$ must have the same size, which implies that $G$ must be $r$-regular. 
Taking $G$ to be the 1-skeleton of the octohedron, $\Delta(G)$ is the matroid independence complex of the uniform matroid $U_{5,6}$ which explains why you were able to find such a representation in this case. More generally, if $G$ is the 1-skeleton of the $n$-dimensional cross polytope then $\Delta(G)$ is isomorphic to the matroid independence complex of $U_{2n-1, 2n}$.
