  In a paper from 1886, Martinetti enumerated small $v_3$-configurations. One
of his tools was a construction that permits to produce a
$(v+1)_3$-configuration from a $v_3$-configuration. He called configurations
that were not constructible in this way irreducible configurations. According
to his definition, the irreducible configurations are Pappus' configuration and
four infinite families of configurations. In 2005, Boben defined a simpler and
more general definition of irreducibility, for which only two
$v_3$-configurations, the Fano plane and Pappus' configuration, remained
irreducible. The present article gives a generalization of Boben's reduction
for both balanced and unbalanced $(v_r,b_k)$-configurations, and proves several
general results on augmentability and reducibility. Motivation for this work is
found, for example, in the counting and enumeration of configurations.
