  We characterize the absolute retracts in the category of reflexive oriented
graphs, that is, antisymmetric reflexive graphs, where morphisms between
objects preserve arcs (which may be sent to loops). Here we show, by correcting
a much earlier attempt at a proof, that a reflexive oriented graph is an
absolute retract if and only if it is indeed a retract of some (direct) product
of reflexive oriented zigzags (which are concatenations of reflexive oriented
paths). Absolute retracts are therefore necessarily acyclic. In contrast to
other categories of graphs and ordered sets, not every acyclic oriented graph
can be embedded isometrically into some absolute retract. Embedding involves
isometry with respect to the zig-zag distances forming a particular "dual
quantale", which is a complete lattice of certain sets of words over the
alphabet $\{+, -\}$, endowed with an additional monoid operation (viz.,
compound concatenation of sets of words) and an involution (interchanging $+$
and $- $ and then mirroring words). As reflexive oriented zigzags have
MacNeille-closed distances, so do their products and retracts. So, the category
of reflexive oriented graphs and its full subcategory of reflexive acyclic
graphs do not have enough injectives, as the injective objects coincide with
the absolute retracts.
