  The goal of this note is to study the action of the backward Rauzy-Veech
algorithm on the translation surfaces with horizontal saddle connections. In
particular, we prove that the orbit of a translation surface via the
aforementioned algorithm is $\infty$-complete if and only if the surface does
not posses horizontal saddle connections. Moreover, we show that appearance of
all saddle connections as sides of the polygonal representations of the
translations surface along the backward Rauzy-Veech induction orbit is
equivalent to the minimality of the horizontal translation flow.
