  Clique-width is a complexity measure of directed as well as undirected
graphs. Rank-width is an equivalent complexity measure for undirected graphs
and has good algorithmic and structural properties. It is in particular related
to the vertex-minor relation. We discuss an extension of the notion of
rank-width to edge-colored graphs. A C-colored graph is a graph where the arcs
are colored with colors from the set C. There is not a natural notion of
rank-width for C-colored graphs. We define two notions of rank-width for them,
both based on a coding of C-colored graphs by edge-colored graphs where each
edge has exactly one color from a field F and named respectively F-rank-width
and F-bi-rank-width. The two notions are equivalent to clique-width. We then
present a notion of vertex-minor for F-colored graphs and prove that F-colored
graphs of bounded F-rank-width are characterised by a finite list of F-colored
graphs to exclude as vertex-minors. A cubic-time algorithm to decide whether a
F-colored graph has F-rank-width (resp. F-bi-rank-width) at most k, for fixed
k, is also given. Graph operations to check MSOL-definable properties on
F-colored graphs of bounded rank-width are presented. A specialisation of all
these notions to (directed) graphs without edge colors is presented, which
shows that our results generalise the ones in undirected graphs.
