Is every distribution that factorizes over a graph a Gibbs field over that graph?

Question

I know that given a graph, a Gibbs field defined over that graph $G = (V,E)$ factorizes with respect to the graph, that is, if $A$, $B$ and $S$ are subsets of vertices such that $S$ separates $A$ and $B$ in the graph, then the random variables $X_A$ and $X_B$ are conditionally independent given $X_S$. Does the converse to this hold? Is every distribution that factorizes over a graph a Gibbs field over that graph? What if we restrict attention to strictly positive distributions?