We tackle the problem of graph transformation with a particular focus on node
cloning. We propose a graph rewriting framework where nodes can be cloned zero,
one or more times. A node can be cloned together with all its incident edges,
with only the outgoing edges, with only the incoming edges or without any of
the incident edges. We thus subsume previous works such as the sesqui-pushout,
the heterogeneous pushout and the adaptive star grammars approaches.

The parameterization process used in the symbolic computation systems Kenzo
and EAT is studied here as a general construction in a categorical framework.
This parameterization process starts from a given specification and builds a
parameterized specification by adding a parameter as a new variable to some
operations. Given a model of the parameterized specification, each
interpretation of the parameter, called an argument, provides a model of the
given specification.