On Reconfiguring Tree Linkages: Trees can Lock

TL;DR

This paper demonstrates that unlike polygons, certain tree linkages in the plane cannot always be reconfigured between simple configurations, revealing exponential complexity in their configuration space.

Contribution

It proves that some trees have exponentially many disconnected configuration classes, showing fundamental differences from polygon reconfiguration.

Findings

Some trees have exponentially many disconnected configuration classes

Tree reconfiguration can be impossible while preserving simplicity

Contrast with polygon reconfiguration results

Abstract

It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex. This result cannot be extended to tree linkages: we show that there are trees with two simple configurations that are not connected by a motion that preserves simplicity throughout the motion. Indeed, we prove that an $N$-link tree can have $2^{\Omega(N)}$ equivalence classes of configurations.