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arXiv · 2505.00163

Tanglegrams with a Unique 1-Crossing-Critical Subtanglegram have Tangle Crossing Number 1

Éva Czabarka·Alec Helm·László Székely
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Abstract

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.

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Éva Czabarka, Alec Helm, László Székely. 2025-04-30. Tanglegrams with a Unique 1-Crossing-Critical Subtanglegram have Tangle Crossing Number 1. https://arxiv.org/abs/2505.00163

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