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A tanglegram consists of two rooted binary trees and a perfect matchingbetween their leaves, and a planar tanglegram is one that admits a layout withno crossings. We show that the problem of generating planar tanglegramsuniformly at random reduces to the corresponding problem for irreducible planartanglegram layouts, which are known to be in bijection with pairs of disjointtriangulations of a convex polygon. We extend the flip operation on a singletriangulation to a flip operation on pairs of disjoint triangulations.Interestingly, the resulting flip graph is both connected and regular, andhence a random walk on this graph converges to the uniform distribution. Wealso show that the restriction of the flip graph to the pairs with a fixedtriangulation in either coordinate is connected, and give diameter bounds thatare near optimal. Our results furthermore yield new insight into the flip graphof triangulations of a convex $n$-gon with a geometric interpretation on theassociahedron.

Sampling planar tanglegrams and pairs of disjoint triangulations