2025/04/30 by Czabarka, Éva, Helm, Alec, Székely, László
#05C05 #05C10 #05C62 #92B10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.00163
A tanglegram of size n is a graph formed from two rooted binary trees with n leaves each and a perfect matching between their leaf sets. Tanglegrams are used to model co-evolution in various settings. A tanglegram layout is a straight line drawing where the two trees are drawn as plane trees with their leaf-sets on two parallel lines, and only the edges of the matching may cross. The tangle crossing number of a tanglegram is the minimum crossing number among its layouts. It is known that tanglegrams have crossing number at least one precisely when they contain one of two size 4 subtanglegrams, which we refer to as cross-inducing subtanglegrams. We show here that a tanglegram with exactly one cross inducing subtanglegram must have tangle crossing number exactly one, and ask the question whether the tangle-crossing number of tanglegrams with exactly k cross-inducing subtanglegrams is bounded for every k.