  A labeled oriented graph (LOG) is an oriented graph with a labeling function
from the edge set into the vertex set. The complexity of a LOG is the minimal
cardinality of an initial set $S$ of vertices such that every vertex can be
reached successively from $S$ only using edges with labels in $S$ or already
visited vertices. We give a constructive proof of a conjecture by Rosebrock
stating that for an interior reduced, connected LOG with $m$ vertices the
complexity is at most $(m+1) / 2$ and show that this bound is sharp.
  Due to results of Howie labeled oriented trees (LOTs) yield crucial
candidates for counterexamples of the Whitehead Conjecture stating that every
subcomplex of an aspherical 2-complex is aspherical. We explicitly describe the
structure of LOTs of maximal complexity $(m+1)/2$. We conclude that the
2-complexes associated to these LOTs are always aspherical excluding them from
the list of possible counterexamples.
