Can every submodular function be represented as a restriction of a graph cut? More precisely, is the following true?
Let $f:2^V \to [0,\infty)$ be submodular, symmetric with $f(\emptyset)=0$. Then, there exist a finite superset $W \supseteq V$ and a weighted graph $G=(W,w)$ with symmetric edge weights $w:W^2 \to [0,\infty)$ such that for all $S \subseteq V$, $$ f(S) = \inf\left\{\sum_{x \in T,y \in W\setminus T} w(x,y) : T \cap V = S, \mbox{ and } T \subseteq W \right\} =:g(S). $$
When restricting to the case $W=V$, then $g(S)$ is just the graph cut, and clearly, not every submodular function can be represented as a graph cut. But what happens when allowing general $W \supseteq V$?
