  The seminal book of Gusfield and Irving [GI89] provides a compact and
algorithmically useful way to represent the collection of stable matches
corresponding to a given set of preferences. In this paper, we reinterpret the
main results of [GI89], giving a new proof of the characterization which is
able to bypass a lot of the "theory building" of the original works. We also
provide a streamlined and efficient way to compute this representation. Our
proofs and algorithms emphasize the connection to well-known properties of the
deferred acceptance algorithm.
