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Unicyclic Networks: Compatibility and Enumeration
January-March 2006 (vol. 3 no. 1)
pp. 84-91
Graphs obtained from a binary leaf labeled ("phylogenetic”) tree by adding an edge so as to introduce a cycle provide a useful representation of hybrid evolution in molecular evolutionary biology. This class of graphs (which we call "unicyclic networks”) also has some attractive combinatorial properties, which we present. We characterize when a set of binary phylogenetic trees is displayed by a unicyclic network in terms of tree rearrangement operations. This leads to a triple-wise compatibility theorem and a simple, fast algorithm to determine 1{\hbox{-}}{\rm cycle} compatibility. We also use generating function techniques to provide closed-form expressions that enumerate unicyclic networks with specified or unspecified cycle length, and we provide an extension to enumerate a class of multicyclic networks.

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Index Terms:
Phylogenetic tree, compatibility, circular orderings, generating function, galled-trees.
Charles Semple, Mike Steel, "Unicyclic Networks: Compatibility and Enumeration," IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 3, no. 1, pp. 84-91, Jan.-March 2006, doi:10.1109/TCBB.2006.14
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