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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.21945 |
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| _version_ | 1866914585652494336 |
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| author | Dai, Shilong Long, Yangjing |
| author_facet | Dai, Shilong Long, Yangjing |
| contents | 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 $μ(B)$ 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 $μ(B)\le k$ 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 $μ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21945 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Characterization of Level-k Realizability for Clustering Systems Dai, Shilong Long, Yangjing Molecular Networks 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 $μ(B)$ 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 $μ(B)\le k$ 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 $μ$. |
| title | A Characterization of Level-k Realizability for Clustering Systems |
| topic | Molecular Networks |
| url | https://arxiv.org/abs/2605.21945 |