most $2(n-1)$ clusters. Thus, for fixed $k$, the size of the input is polynomial in $n$. It follows from these observations that whether a set of clusters is represented by a given simple level-$k$ network can be checked in polynomial time.

![The single level-1 generator and the four level-2 generators. Here the sides have been labelled with capital letters.[]{data-label="fig:gen"}](./Figures/VanIersel34fig7)

![image](./Figures/exLemma1)

It is known that, if the leaves of $N$ are removed and all nodes with both indegree and outdegree equal to 1 are suppressed, the resulting structure will be a level-$k$ generator, defined in [@lev2TCBB]. See also Figure \[fig:gen\]. For fixed $k$, there are only a constant number of level-$k$ generators [@Gambette2009structure Proposition 2.5]. Recall that the *sides* of a level-$k$ generator are defined as the union of its edges and its nodes of indegree-2 and outdegree-0. For fixed $k$ the maximum number of sides ranging over all level-$k$ generators, is a constant.

For a cluster set ${{\mathcal C}}$ on ${{\mathcal X}}$, we write $x \rightarrow y$ if and only if every non-singleton cluster in ${{\mathcal C}}$ that contains $x$, also contains $y$. For example, for the cluster set of Figure \[fig:exLemma1\], we have $e \rightarrow d$ and $f \rightarrow 