A Locked Orthogonal Tree

TL;DR

This paper presents a counterexample to a conjecture about flattening orthogonal trees in 2D without self-intersection, introducing the smallest known locked tree with only eleven edges.

Contribution

It extends previous results on locked linkages to include zero-length edges, providing the first counterexample to the conjecture and identifying the smallest locked tree.

Findings

Counterexample to Poon's conjecture on flattening orthogonal trees

Extension of locked linkage results to zero-length edges

Smallest known locked tree with eleven edges

Abstract

We give a counterexample to a conjecture of Poon [Poo06] that any orthogonal tree in two dimensions can always be flattened by a continuous motion that preserves edge lengths and avoids self-intersection. We show our example is locked by extending results on strongly locked self-touching linkages due to Connelly, Demaine and Rote [CDR02] to allow zero-length edges as defined in [ADG07], which may be of independent interest. Our results also yield a locked tree with only eleven edges, which is the smallest known example of a locked tree.