Assume we have a surface $S$ (smooth if you want), and a map $f: S \to V$ that contracts a curve $C \subset S$. What condition would give a factoring of $f$ through a contraction $c: S \to V'$ contracting $C$ but otherwise being an isomorphism?

What about the special case where $S$ is a flat family of curves and $C$ is the central fiber (all other fibers are assumed smooth). What criterion would give a factoring of $f$ through the contraction?