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The fundamental group of the complement of a plane curve is a very importanttopological invariant. In particular, it is interesting to find out whetherthis group is determined by the combinatorics of the curve or not, and whetherit is a direct sum of free groups and a free abelian group, or it has aconjugation-free geometric presentation.  In this paper, we investigate the structure of this fundamental group whenthe graph of the conic-line arrangement is a unique cycle of length $n$ and theconic passes through all the multiple points of the cycle. We show that if n isodd, then the affine fundamental group is abelian but not conjugation-free. Forthe even case, if n>4, then using quotients of the lower central series, weshow that the fundamental group is not even a direct sum of a free abeliangroup and free groups.

On the structure of fundamental groups of conic-line arrangements having a cycle in their graph