A note on reconfiguring tree linkages: trees can lock

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 Ω(N) equivalence classes of configurations.

Folding Equilateral Plane Graphs

“…Biedl et al [2] introduced the notion of a "locked tree" and gave the first example thereof. Here a tree refers to a plane tree linkage, that is, a tree graph with specified edge lengths and a preferred planar embedding.…”

“…Such a linkage can move or fold continuously subject to the constraints that the edges remain straight line segments of the specified lengths, and that the edges never properly cross each other [3]. A tree is universally foldable [2] if it can be folded continuously from any configuration to any other. Equivalently, a tree is universally foldable if it can be folded from any configuration into a canonical configuration, in which the edges lie along a horizontal line and point rightward from a single root vertex.…”

“…Otherwise, a tree is locked . The construction in [2] gives a specific tree configuration that is locked in the sense that it cannot be folded to a canonical configuration.…”

“…Note that Ballinger et al [1] call this a "flat configuration"; we use the term "canonical configuration" instead to minimize the ambiguity of the word "flat." It is useful to interpret canonical states of trees as "unfolded" states, because all canonical states are equivalent: for any desired vertex v and edge e incident to v , there exists a continuous motion from any canonical state to the canonical state rooted at v in which edge e is the topmost edge incident to v [2].…”

“…By [2], we may move H k to a new canonical state with w 2 at the root, and (w 2 , w 1 ) as the topmost edge incident to w 2 (thereby making (w 2 , w 4 ) the bottommost edge incident to w 2 ). This motion of H k to a new state N k induces a motion of G to a new state M k = g * k (N k ) lying on a horizontal line.…”