  A subgraph H= (V, F) of a graph G= (V,E) is non-separating if G-F, that is,
the graph obtained from G by deleting the edges in F, is connected. Analogously
we say that a subdigraph X= (V,B) of a digraph D= (V,A) is non-separating if
D-B is strongly connected. We study non-separating spanning trees and
out-branchings in digraphs of independence number 2. Our main results are that
every 2-arc-strong digraph D of independence number alpha(D) = 2 and minimum
in-degree at least 5 and every 2-arc-strong oriented graph with alpha(D) = 2
and minimum in-degree at least 3 has a non-separating out-branching and minimum
in-degree 2 is not enough. We also prove a number of other results, including
that every 2-arc-strong digraph D with alpha(D)<=2 and at least 14 vertices has
a non-separating spanning tree and that every graph G with delta(G)>=4 and
alpha(G) = 2 has a non-separating hamiltonian path.
