  We address the problem of decomposing graphs in perturbative QCD into terms
associated with particular regions. Motivated by asking how to incorporate
next-to-leading order (NLO) QCD corrections in parton-shower algorithms, we
require that: (a) The integrand for the hard part is to be integrable even if
the corrections are applied to a process that is not infrared and collinear
safe. (b) The splitting between the terms should be defined gauge-invariantly.
(c) The dependence on cut-offs should obey homogeneous evolution equations. In
the context of one-gluon-emission graphs for deep inelastic scattering, we
explain a subtractive technique that is based on gauge-invariant Wilson-line
operators. Appropriate organization of subtractions involving the soft region
allows a connection to previous work where evolution equations with respect to
the directions of the Wilson lines have been derived.
