  Graph minors are a primary tool in understanding the structure of undirected
graphs, with many conceptual and algorithmic implications. We propose new
variants of \emph{directed graph minors} and \emph{directed graph embeddings},
by modifying familiar definitions. For the class of 2-terminal directed acyclic
graphs (TDAGs) our two definitions coincide, and the class is closed under both
operations. The usefulness of our directed minor operations is demonstrated by
characterizing all TDAGs with serial-parallel width at most $k$; a class of
networks known to guarantee bounded negative externality in nonatomic routing
games. Our characterization implies that a TDAG has serial-parallel width of
$1$ if and only if it is a directed series-parallel graph. We also study the
computational complexity of finding a directed minor and computing the
serial-parallel width.
