Extending some induced substructures of an inversive plane

Abstract

Given a circle $$C$$C of an inversive plane $${\mathcal {I}}$$I of order $$n$$n, the remaining circles are partitioned into three types according to the number of intersection points with $$C$$C. Let $${\mathcal {S}}$$S be the incidence structure formed by the points of $${\mathcal {I}}$$I and any two types of circles. It is proved that with some additional requirements, $${\mathcal {I}}$$I is the only inversive plane of order $$n$$n having $${\mathcal {S}}$$S as an induced substructure.