At least \(n - 1\) intersection points in a connected family of \(n\) unit circles in the plane (Q2464365)

In [Geom. Dedicata 33, 227--238 (1990; Zbl 0699.52003)] \textit{A.~Bezdek} stated the following beautiful conjecture: given \(n\) unit circles in the plane, such that their union is a connected set, then the number of intersection points determined by the boundary of the circles is greater than or equal to \(n-1\); equality holds only for some particular tree arrangements which were also described in the paper. In the special case when no two circles touch it was already shown by K.~Bezdek and R.~Connelly in 1988 that there are at least \(n\) intersection points. In this paper the authors confirm the first part of A.~Bezdek's conjecture; the second part, regarding the equality case, is still open.