Reconstruction of caterpillar tanglegrams

TL;DR

This paper investigates whether a caterpillar tanglegram can be uniquely reconstructed from the multiset of all its size-$(n-1)$ induced tanglegrams, providing an affirmative answer under certain conditions.

Contribution

It proves that a caterpillar tanglegram is uniquely determined by its induced size-$(n-1)$ tanglegrams, a novel result in tanglegram reconstruction.

Findings

Unique reconstruction for caterpillar tanglegrams established

Multiset of size-$(n-1)$ tanglegrams suffices for reconstruction

Provides theoretical foundation for tanglegram analysis

Abstract

A tanglegram consists of two rooted binary trees with the same number of leaves and a perfect matching between the leaves of the trees. Given a size-$n$ tanglegram, i.e., a tanglegram for two trees with $n$ leaves, a multiset of induced size-$(n-1)$ tanglegrams is obtained by deleting a pair of matched leaves in every possible way. Here, we analyze whether a size-$n$ tanglegram is uniquely encoded by this multiset of size-$(n-1)$ tanglegrams. We answer this question affirmatively in the case that at least one of the two trees of the tanglegram is a caterpillar tree.