  We address the problem of cyclic termgraph rewriting. We propose a new
framework where rewrite rules are tuples of the form $(L,R,\tau,\sigma)$ such
that $L$ and $R$ are termgraphs representing the left-hand and the right-hand
sides of the rule, $\tau$ is a mapping from the nodes of $L$ to those of $R$
and $\sigma$ is a partial function from nodes of $R$ to nodes of $L$. $\tau$
describes how incident edges of the nodes in $L$ are connected in $R$. $\tau$
is not required to be a graph morphism as in classical algebraic approaches of
graph transformation. The role of $\sigma$ is to indicate the parts of $L$ to
be cloned (copied). Furthermore, we introduce a new notion of \emph{cloning
pushout} and define rewrite steps as cloning pushouts in a given category.
Among the features of the proposed rewrite systems, we quote the ability to
perform local and global redirection of pointers, addition and deletion of
nodes as well as cloning and collapsing substructures.
