  In this paper, we build on recent results by Chauve et al. (2014) and Bahrani
and Lumbroso (2017), which combined the split-decomposition, as exposed by
Gioan and Paul, with analytic combinatorics, to produce new enumerative results
on graphs---in particular the enumeration of several subclasses of perfect
graphs (distance-hereditary, 3-leaf power, ptolemaic). Our goal was to study a
simple family of graphs, of which the split-decomposition trees have prime
nodes drawn from an enumerable (and manageable!) set of graphs. Cactus graphs,
which we describe in more detail further down in this paper, can be thought of
as trees with their edges replaced by cycles (of arbitrary lengths). Their
split-decomposition trees contain prime nodes that are cycles, making them
ideal to study. We derive a characterization for the split-decomposition trees
of cactus graphs, produce a general template of symbolic grammars for cactus
graphs, and implement random generation for these graphs, building on work by
Iriza (2015).
