  A cyclic presentation of a group is a presentation with an equal number of
generators and relators that admits a particular cyclic symmetry. We
characterise the orientable, non-orientable, and redundant cyclic presentations
and obtain concise refinements of these presentations. We show that the Tits
alternative holds for the class of groups defined by redundant cyclic
presentations and that if the number of generators of the cyclic presentation
is greater than two then the corresponding group is large. Generalizing and
extending earlier results of the authors we describe the star graphs of
orientable and non-orientable cyclic presentations and classify the cyclic
presentations whose star graph components are pairwise isomorphic incidence
graphs of generalized polygons, thus classifying the so-called
$(m,k,\nu)$-special cyclic presentations.
