Abstract
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 $\frac{1}{2}\binom{n}{2}(1-o(1))\le\gamma(n)<\frac{1}{2}\binom{n}{2}$. For an arbitrary tanglegram $T$, the tangle crossing number, $\mathrm{crt}(T)$, is NP-hard to compute (Fernau et al. 2005). We provide an algorithm which lower bounds $\mathrm{crt}(T)$ 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).
Highlights
A drawing D(G) of a graph G in the plane is a set of distinct points in the plane, one for each vertex of G, and a collection of simple open arcs, one for each edge of the graph, such that if e is an edge of G with endpoints v and w, the closure of the arc α representing e consists precisely of α and the two points representing v and w.We further require that no edge–arc intersects any vertex point
We find that the behavior is quite similar for the tangle crossing number
For each n 4, we define a tanglegram of size n with tangle crossing number n−3 for which there is a single matching edge whose removal yields a planar subtanglegram
Summary
A drawing D(G) of a graph G in the plane is a set of distinct points in the plane, one for each vertex of G, and a collection of simple open arcs, one for each edge of the graph, such that if e is an edge of G with endpoints v and w, the closure (in the plane) of the arc α representing e consists precisely of α and the two points representing v and w. For each k 1, they define a 1-edge planar graph Gk with 2k + 4 vertices, 6k + 7 edges, and crossing number k. For each n 4, we define a tanglegram of size n with tangle crossing number n−3 for which there is a single matching edge whose removal yields a planar subtanglegram. We examine the largest tangle crossing number of a tanglegram of size n (an analogue of the crossing number of the complete graph on n vertices). Drawing random tanglegrams of size n from a uniform distribution, we give computational evidence that these lower bounds are Θ(n2) with high probability, matching the result of Czabarka, Szekely, Wagner [5] that such a tanglegram has tangle crossing number Θ(n2) with high probability
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