Circular-arc graph for distinct distances

Around each of $n$ points draw its at most $t$ distinct-distance circles. On every circle containing at least three points, join cyclically consecutive points by arcs. The resulting loopless multigraph has at least $n(n-1)-2nt$ edges and a drawing with $O(n^2{t}^2)$ crossings. A pair joined by several arcs has every corresponding centre on its perpendicular bisector, which is the key to bounding multiplicity.

Every supporting centre of an arc between fixed endpoints lies on their perpendicular bisector. Multiplicity at least $k$ therefore makes that line contain at least $k$ of the original points. For each such line containing $s$ points, its $s$ centres each support at most $t$ circles; on one circle at most two consecutive-point arcs have that symmetry axis. The rich line bound summed over dyadic occupancies gives the displayed bound.