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The fundamental group of the complement of a hyperplane arrangement plays animportant role in studying the corresponding arrangements. In particular, forlarge families of hyperplane arrangements, this fundamental group, beingisomorphic to the fundamental group of a complement of a line arrangement, hassome remarkable properties: either it is a direct sum of free groups and a freeabelian group, or it has a conjugation-free geometric presentation.  In this paper, we first give a complete proof to the following key lemma: ifwe draw a new line through only one intersection point of a given real linearrangement whose fundamental group is conjugation-free, then the fundamentalgroup of the new arrangement is also conjugation-free.  Second, we generalize this lemma to the case of conic-line arrangements.Moreover, we prove that once the graph associated to conic-line arrangements(defined slightly different than the corresponding graph for line arrangements)has no cycles, then the fundamental group of its complement has aconjugation-free geometric presentation and in addition can be written as adirect sum of free groups and a free abelian group. Also, we show that if thegraph consists of one cycle, and the conic does not pass through all themultiple points corresponding to the vertices of the cycle, then thefundamental group has a conjugation-free geometric presentation as well.  For conclusion, we extend the family of real line arrangements having aconjugation-free geometric presentation (for their fundamental group) bydefining the notion of a conjugation-free graph. We also extend this notion tocertain families of conic-line arrangements.

On the structure of conjugation-free fundamental groups of conic-line arrangements