A Characterization of Level-k Realizability for Clustering Systems

We give a Hasse-diagram characterization of when a clustering system $\mathcal C$ on a finite taxa set $X$ is the hardwired clustering system $C_N$ of a rooted level-$k$ network. For each non-trivial block $B$ of $H=\mathcal H[\mathcal C]$, we define a parameter $\mu(B)$ using minimum families of clusters that generate all overlap-intersections inside $B$. The main theorem proves that there exists a rooted level-$k$ network $N$ with $C_N=\mathcal C$ if and only if $\mu(B)\le k$ for every non-trivial block $B$ of $H$. The necessity proof shows that overlap-intersection pieces must be represented by non-root hybrid vertices in any realizing block. The sufficiency proof is constructive: starting from the Hasse diagram, it iteratively splits selected hybrid vertices, preserves the hardwired clustering system, and terminates with a realization whose level is bounded by the block-wise values of $\mu$.