Non-separating Spanning Trees and Out-Branchings in Digraphs of Independence Number 2

A subgraph H= (V, E^′) of a graph G= (V, E) is non-separating if G\ E^′, that is, the graph obtained from G by deleting the edges in E^′, is connected. Analogously we say that a subdigraph X= (V, A^′) of a digraph D= (V, A) is non-separating if D\ A^′ 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 α(D) = 2 and minimum in-degree at least 5 and every 2-arc-strong oriented graph with α(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 α(D) ≤ 2 and at least 14 vertices has a non-separating spanning tree and that every graph G with δ(G) ≥ 4 and α(G) = 2 has a non-separating Hamiltonian path.