Near polygons from partial linear spaces
Starting from a partial linear space \({\mathcal S} =(P, {\mathcal L})\) the authors discuss four constructions which alter the structure of \({\mathcal S}\), thus obtaining infinitely many non-degenerate as well as degenerate near polygons. An incidence structure is called a partial linear space, if every line is incident with at least two points and two distinct points lie on at most one line. Such a space is called degenerate, if there is a point incident with exactly one point. A partial linear space is said to be a near polygon, if the diameter of its point graph \(\Gamma\) is finite and if for every pair \((p,L) \in P \times {\mathcal L}\) there is a unique point \(q \in P\) on \(L\) nearest to \(p\) with respect to the natural distance defined on \(\Gamma\). Using their constructions the authors characterize those near polygons which contain a point that is of distance 2 from any other point. The paper closes with a characterization of Hamming near polygons, i.e. polygons that can be written as a direct product of lines, in terms of parallelism.
scientific article; zbMATH DE number 1334361