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A conjugation-free geometric presentation of a fundamental group is apresentation with the natural topological generators $x_1, ..., x_n$ and thecyclic 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 thegenerators.  We have already proved that if the graph of the arrangement is a disjointunion of cycles, then its fundamental group has a conjugation-free geometricpresentation. In this paper, we extend this property to arrangements whosegraphs are a disjoint union of cycle-tree graphs.  Moreover, we study some properties of this type of presentations for afundamental group of a line arrangement's complement. We show that thesepresentations satisfy a completeness property in the sense of Dehornoy, if thecorresponding graph of the arrangement has no edges. The completeness propertyis a powerful property which leads to many nice properties concerning thepresentation (such as the left-cancellativity of the associated monoid andyields some simple criterion for the solvability of the word problem in thegroup).

A conjugation-free geometric presentation of fundamental groups of arrangements II: Expansion and some properties