  We describe a kernel of size 9k-8 for the NP-hard problem of computing the
Tree Bisection and Reconnect (TBR) distance k between two unrooted binary
phylogenetic trees. We achieve this by extending the existing portfolio of
reduction rules with three novel new reduction rules. Two of the rules are
based on the idea of topologically transforming the trees in a
distance-preserving way in order to guarantee execution of earlier reduction
rules. The third rule extends the local neighbourhood approach introduced in
(Kelk and Linz, Annals of Combinatorics 24(3), 2020) to more global structures,
allowing new situations to be identified when deletion of a leaf definitely
reduces the TBR distance by one. The bound on the kernel size is tight up to an
additive term. Our results also apply to the equivalent problem of computing a
Maximum Agreement Forest (MAF) between two unrooted binary phylogenetic trees.
We anticipate that our results will be more widely applicable for computing
agreement-forest based dissimilarity measures.
