Rooted NNI moves and distance-1 tail moves on tree-based phylogenetic networks

Abstract We show that any two tree-based rooted binary phylogenetic networks (with the same number of reticulation nodes and identical degrees at the root) are connected by at most ( 4 k + 8 ) ⋅ | X | + 3 4 k + | X | ⋅ ⌈ log 2 | X | ⌉ rooted nearest-neighbour interchange ( r N N I ) moves. As a corollary of the proof we obtain that the diameter is at most a 2 k additive term larger for distance-1 tail moves.