  In this paper we introduce the linear clique-width, linear NLC-width,
neighbourhood-width, and linear rank-width for directed graphs. We compare
these parameters with each other as well as with the previously defined
parameters directed path-width and directed cut-width. It turns out that the
parameters directed linear clique-width, directed linear NLC-width, directed
neighbourhood-width, and directed linear rank-width are equivalent in that
sense, that all of these parameters can be upper bounded by each of the others.
For the restriction to digraphs of bounded vertex degree directed path-width
and directed cut-width are equivalent. Further for the restriction to
semicomplete digraphs of bounded vertex degree all six mentioned width
parameters are equivalent. We also show close relations of the measures to
their undirected versions of the underlying undirected graphs, which allow us
to show the hardness of computing the considered linear width parameters for
directed graphs. Further we give first characterizations for directed graphs
defined by parameters of small width.
