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Issue No. 01 - January-March (2006 vol. 3)
ISSN: 1545-5963
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.
Phylogenetic tree, compatibility, circular orderings, generating function, galled-trees.

M. Steel and C. Semple, "Unicyclic Networks: Compatibility and Enumeration," in IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 3, no. , pp. 84-91, 2006.
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