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The authors establish a coordinate-based characterization of tree-child phylogenetic networks using $\mu$-representations in $\mathbb{N}^n$. By connecting ordered tree-path graph decompositions directly to vectorial properties, they prove that a finite set of integer vectors corresponds to a valid tree-child network if and only if it satisfies tree-child $\mu$-compatibility. This provides an exact feasibility criterion that bypasses topological graph traversal for network validation and structural inference.
Complex reticulate network topologies can now be verified entirely through integer vectors, establishing an exact if-and-only-if algebraic test for tree-child phylogenetic reconstruction.
We characterize tree-child phylogenetic networks in terms of their mu-representations. First, we give a structural characterization of tree-child networks by means of ordered tree-path decompositions. We then translate this decomposition into a set of purely vectorial conditions on finite subsets M in N^n. We prove that such a set M is the mu-representation of a tree-child phylogenetic network if and only if it is tree-child mu-compatible. This provides a feasibility criterion for tree-child mu-representations which can be used as a basis for reconstruction and further algorithmic applications. Note that this paper presents results arising from ongoing research on tree-child networks and that the results will be further developed and placed into proper context in subsequent versions.