  Chudnovsky and Seymour's structure theorem for claw-free graphs has led to a
multitude of recent results that exploit two structural operations: {\em
compositions of strips} and {\em thickenings}. In this paper we consider the
latter, proving that every claw-free graph has a unique optimal {\em
antithickening}, where our definition of {\em optimal} is chosen carefully to
respect the structural foundation of the graph. Furthermore, we give an
algorithm to find the optimal antithickening in $O(m^2)$ time. For the sake of
both completeness and ease of proof, we prove stronger results in the more
general setting of trigraphs.
