  Forbidden characterizations may sometimes be the most natural way to describe
families of graphs, and yet these characterizations are usually very hard to
exploit for enumerative purposes.
  By building on the work of Gioan and Paul (2012) and Chauve et al. (2014), we
show a methodology by which we constrain a split-decomposition tree to avoid
certain patterns, thereby avoiding the corresponding induced subgraphs in the
original graph.
  We thus provide the grammars and full enumeration for a wide set of graph
classes: ptolemaic, block, and variants of cactus graphs (2,3-cacti, 3-cacti
and 4-cacti). In certain cases, no enumeration was known (ptolemaic, 4-cacti);
in other cases, although the enumerations were known, an abundant potential is
unlocked by the grammars we provide (in terms of asymptotic analysis, random
generation, and parameter analyses, etc.).
  We believe this methodology here shows its potential; the natural next step
to develop its reach would be to study split-decomposition trees which contain
certain prime nodes. This will be the object of future work.
