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

TL;DR

This work studies tanglegrams and their crossing behavior, focusing on the scenario where a tanglegram contains exactly one cross-responsible subtanglegram, i.e., a unique copy of a size-4 cross-inducing configuration. It develops a scar-based framework and analyzes induced subtanglegrams corresponding to the two 1-crossing-critical cases, $\mathcal{K}_1$ and $\mathcal{K}_2$, to bound the tangle crossing number. The main results show that any tanglegram with a single cross-responsible set satisfies $\operatorname{crt}(\mathcal{T})=1$, providing a step toward a full classification of 2-crossing-critical tanglegrams and connecting cross-responsible sets to crossing behavior via a tanglegram Kuratowski analogue. The paper concludes with open questions on whether $\operatorname{crt}(\mathcal{T})$ remains bounded for tanglegrams with exactly $k$ cross-responsible sets, highlighting future directions in this line of inquiry.

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.

Figures (9)

• Figure 1: The 1-crossing-critical tanglegrams $\mathcal{K}_1$ and $\mathcal{K}_2$.
• Figure 2: A tanglegram that contains a subdivided $K_{3,3}$ such that all vertices of the $K_{3,3}$ are in the left tree. Vertex classes are marked with numbers vs. letters, subdivided edges are bold.
• Figure 3: The gray shaded regions represent a planar tanglegram $F_m$ with $m$ matching edges drawn in a planar way. If two shaded regions cross, all matching edges between the two copies cross. The tanglegram $T_1$ on the left has a single cross-responsible set that induces a $\mathcal{K}_1$, but its associated graph $T_1^*$ has $m+1$ subdivided $K_{3,3}$-s: the cross-responsible set with any matching edge from $F_m$ induces a subtanglegram in $T_1$ that is a subdivision of the graph $\mathcal{K}_1^*$. The middle picture is an optimal layout of tanglegram $T_2$ with tangle crossing number $m^2$, and the rightmost picture is an optimal rectilinear drawing of its associated graph $T_2^*$ with exactly one crossing.
• Figure 4: Example for Lemma \ref{['lm:k1notinside']}: adding the edge with scars at $p$ and $q$ (dashed line) and removing the matching edge on the length $3$ path connecting $v_1$ and $v_2$ results in a copy of $\mathcal{K}_1$.
• Figure 5: Illustration for Theorem \ref{['th:k1']}.

Theorems & Definitions (34)

• Theorem 1
• Theorem 2
• Definition 1
• Definition 2
• Definition 3
• Lemma 3
• proof
• Definition 4
• Definition 5
• Definition 6