  Stable marriage of a two-sided market with unit demand is a classic problem
that arises in many real-world scenarios. In addition, a unique stable marriage
in this market simplifies a host of downstream desiderata. In this paper, we
explore a new set of sufficient conditions for unique stable matching (USM)
under this setup. Unlike other approaches that also address this question using
the structure of preference profiles, we use an algorithmic viewpoint and
investigate if this question can be answered using the lens of the deferred
acceptance (DA) algorithm (Gale and Shapley, 1962). Our results yield a set of
sufficient conditions for USM (viz., MaxProp and MaxRou) and show that these
are disjoint from the previously known sufficiency conditions like sequential
preference and no crossing. We also provide a characterization of MaxProp that
makes it efficiently verifiable, and shows the gap between MaxProp and the
entire USM class. These results give a more detailed view of the sub-structures
of the USM class.
