  A binary phylogenetic network may or may not be obtainable from a tree by the
addition of directed edges (arcs) between tree arcs. Here, we establish a
precise and easily tested criterion (based on `2-SAT') that efficiently
determines whether or not any given network can be realized in this way.
Moreover, the proof provides a polynomial-time algorithm for finding one or
more trees (when they exist) on which the network can be based. A number of
interesting consequences are presented as corollaries; these lead to some
further relevant questions and observations, which we outline in the
conclusion.
