  In this paper we establish a new characterisation of star-regular categories,
using a property of internal reflexive graphs, which is suggested by a recent
result due to O. Ngaha Ngaha and the first author. We show that this property
is, in a suitable sense, invariant under regular completion of a category in
the sense of A. Carboni and E. M. Vitale. Restricting to pointed categories,
where star-regularity becomes normality in the sense of the second author, this
reveals an unusual behaviour of the exactness property of normality (i.e. the
property that regular epimorphisms are normal epimorphisms) compared to other
closely related exactness properties studied in categorical algebra.
