Sorry for the somewhat vague question - I am pretty unfamiliar with 3-manifolds.
Let $M$ be a 3-manifold and let $S$ be a properly embedded surface compact surface in $M$. Is there a notion of "fully compressing $S$" - I imagine that this involves taking a maximal collection of disjoint compressing disks for $S$ and compressing along all of them. How does one make "maximal" precise? And is the resulting surface $S'$ unique in any way (i.e. independent of the choice of the compressing disks)? A reference for these things would be great.