Tanglegrams are special graphs that consist of a pair of rooted binary trees
with the same number of leaves, and a perfect matching between the two
leaf-sets. These objects are of use in phylogenetics and are represented with
straightline drawings where the leaves of the two plane binary trees are on two
parallel lines and only the matching edges can cross. The tangle crossing
number of a tanglegram is the minimum crossing number over all such drawings
and is related to biologically relevant quantities, such as the number of times
a parasite switched hosts.
Our main results for tanglegrams which parallel known theorems for crossing
numbers are as follows. The removal of a single matching edge in a tanglegram
with n leaves decreases the tangle crossing number by at most n−3, and this
is sharp. Additionally, if γ(n) is the maximum tangle crossing number of
a tanglegram with n leaves, we prove
21(2n)(1−o(1))≤γ(n)<21(2n). Further,
we provide an algorithm for computing non-trivial lower bounds on the tangle
crossing number in O(n4) time. This lower bound may be tight, even for
tanglegrams with tangle crossing number Θ(n2).Comment: 13 pages, 6 figure