Analogies between the Crossing Number and the Tangle Crossing Number

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 straight-line 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 number of crossings 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 gamma(n) is the maximum tangle crossing number of a tanglegram with n leaves, we prove 1/2 ((n)(2)) (1-o(1)) <= gamma(n) < 1/2 ((n)(2)). For an arbitrary tanglegram T, the tangle crossing number, crt(T), is NP-hard to compute (Fernau et al. 2005). We provide an algorithm which lower bounds crt(T) and runs in O(n(4)) time. To demonstrate the strength of the algorithm, simulations on tanglegrams chosen uniformly at random suggest that the tangle crossing number is at least 0.055n(2) with high probabilty, which matches the result that the tangle crossing number is Theta(n(2)) with high probability (Czabarka et al. 2017).