A Tight Kernel for Computing the Tree Bisection and Reconnection Distance between Two Phylogenetic Trees

Steven Kelk*, Simone Linz

*Corresponding author for this work

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In 2001 Allen and Steel showed that, if subtree and chain reduction rules have been applied to two unrooted phylogenetic trees, the reduced trees will have at most 28k taxa where k is the tree bisection and reconnection distance between the two trees. Here we reanalyze Allen and Steel's kernelization algorithm and prove that the reduced instances will in fact have at most 15k - 9 taxa. Moreover we show, by describing a family of instances which have exactly 15k - 9 taxa after reduction, that this new bound is tight. These instances also have no common clusters, showing that a third commonly encountered reduction rule, the cluster reduction, cannot further reduce the size of the kernel in the worst case. To achieve these results we introduce and use "unrooted generators" which are analogues of rooted structures that have appeared earlier in the phylogenetic networks literature. Using similar arguments we show that, for the minimum hybridization problem on two rooted trees, 9k - 2 is a tight bound (when subtree and chain reduction rules have been applied) and 9k - 4 is a tight bound (when, additionally, the cluster reduction has been applied) on the number of taxa, where k is the hybridization number of the two trees.

Original languageEnglish
Pages (from-to)1556-1574
Number of pages19
JournalSiam Journal on Discrete Mathematics
Issue number3
Publication statusPublished - 2019


  • fixed-parameter tractability
  • tree bisection and reconnection
  • generator
  • kernelization
  • phylogenetic network
  • phylogenetic tree
  • hybridization number

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