  A conjugation-free geometric presentation of a fundamental group is a
presentation with the natural topological generators $x_1, ..., x_n$ and the
cyclic relations: $x_{i_k}x_{i_{k-1}} ... x_{i_1} = x_{i_{k-1}} ... x_{i_1}
x_{i_k} = ... = x_{i_1} x_{i_k} ... x_{i_2}$ with no conjugations on the
generators.
  We have already proved that if the graph of the arrangement is a disjoint
union of cycles, then its fundamental group has a conjugation-free geometric
presentation. In this paper, we extend this property to arrangements whose
graphs are a disjoint union of cycle-tree graphs.
  Moreover, we study some properties of this type of presentations for a
fundamental group of a line arrangement's complement. We show that these
presentations satisfy a completeness property in the sense of Dehornoy, if the
corresponding graph of the arrangement has no edges. The completeness property
is a powerful property which leads to many nice properties concerning the
presentation (such as the left-cancellativity of the associated monoid and
yields some simple criterion for the solvability of the word problem in the
group).
