  In the context of integrable partial difference equations on quad-graphs, we
introduce the notion of open boundary reductions as a new means to construct
discrete integrable mappings and their invariants. This represents an
alternative to the well-known periodic reductions. The construction deals with
well-posed initial value problems for quad equations on quad-graphs restricted
to a strip. It relies on the so-called double-row monodromy matrix and gives
rise to integrable mappings. To obtain the double-row monodromy matrix, we use
the notion of boundary matrix and discrete boundary zero curvature condition,
themselves related to the boundary consistency condition, which complements the
well-known $3$D consistency condition for integrable quad equations and gives
an integrability criterion for boundary equations. This relation is made
precise in this paper. Our focus is on quad equations from the
Adler-Bobenko-Suris (ABS) classification and their discrete integrable boundary
equations. Taking as our prime example a regular $\mathbb{Z}^2$-lattice with
two parallel boundaries, we provide an explicit construction of the maps
obtained by open boundary reductions, the boundary matrices, as well as the
invariants extracted from the double-row monodromy matrix. Examples of
integrable maps are considered for the H1 and Q1($\delta =0$) equations and an
interesting example of a non-QRT map of the plane is presented. Examples of
well-posed quad-graph systems on a strip beyond the $\mathbb{Z}^2$-lattice are
also given.
