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To resolve the visualization and structural complexity of reticulate evolutionary structures, this paper establishes rigorous planarity criteria for galled phylogenetic networks. The authors prove that any planar galled network is necessarily upward planar, ensuring directed hierarchy can be preserved without introducing edge crossings. Furthermore, they provide exact tripartite characterizations for outerplanar and terminal planar galled networks based on forbidden vertex configurations, forbidden directed subgraphs, and forbidden undirected minors.
Planar galled networks are fundamentally upward planar, resolving layout ambiguities and establishing exact forbidden-minor boundaries for crossing-free hierarchical DAGs.
Rooted phylogenetic networks are widely used to represent the evolution of species that have undergone reticulate processes. However, these networks can be highly non-planar, making them more difficult to visualise and interpret than evolutionary trees. In this paper, we investigate planarity properties of galled networks, an important subclass of phylogenetic networks. We show that all planar galled networks are necessarily upward planar. Furthermore, by leveraging recent results on planar phylogenetic networks, we provide three characterisations for each of the outerplanar and terminal planar galled network classes in terms of forbidden vertex configurations, forbidden directed subgraphs, and forbidden structures in their associated underlying undirected graphs. These results contribute to a deeper understanding of the structural properties of galled networks and may inform future methods for their construction and visualisation.
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