On Straightening Low-Diameter Unit Trees

A polygonal chain is a sequence of consecutively joined edges embedded in space. A k-chain is a chain of k edges. A polygonal tree is a set of edges joined into a tree structure embedded in space. A unit tree is a tree with only edges of unit length. A chain or a tree is simple if non-adjacent edges do not intersect. We consider the problem about the reconfiguration of a simple chain or tree through a series of continuous motions such that the lengths of all tree edges are preserved and no edge crossings are allowed. A chain or tree can be straightened if all its edges can be aligned along a common straight line such that each edge points “away” from a designed leaf node. Otherwise it is called locked. Graph reconfiguration problems have wide applications in contexts including robotics, molecular conformation, rigidity and knot theory. The motivation for us to study unit trees is that for instance, the bonding-lengths in molecules are often similar, as are the segments of robot arms. A chain in 2D can always be straightened [4, 5]. In 4D or higher, a tree can always be straightened [3]. There exist trees [2] in 2D and 5-chains in 3D that can lock. Alt et al. [1] showed that deciding the reconfigurability for trees in 2D and for chains in 3D is PSPACE-complete. However the problem of deciding straightenability for trees in 2D and for chains in 3D remains open. It is easy to verify that a tree of diameter at most 3 in 2D or 3D can always be straightened. In this paper, we show that some tree of diameter 4 in 2D or 3D can lock, and a unit tree of diameter 4 in 2D can always be straightened. In 2D, even a tree with diameter as low as 6 can lock [2] as shown in Figure 1 (a). We present a locked tree of diameter 4 in Figure 1 (b), which simulates the tree in (a). It can be shown locked using the same technique as the proof for (a) by assigning the corresponding equilibrium stresses to the tree edges. In 3D, a 5-chain can lock [2]. We present a 3D locked tree of diameter 4, which is shown in Figure 1 (c). We now consider the straightenability of a unit tree T of diameter 4 in 2D. The center of tree T , denoted by o, is the middle vertex of any 4-chain in T . We call a path connecting the center to a leaf a branch of T . A direct straightening of branch B = ouv in T means to rotate v around u until ouv is straightened by passing through the smaller angle. We denote the sweeping region for directly straightening B by S(B). The direct straightening of B is interfered by another branch B if S(B) ∩ B 6= ∅. There are two kinds of interferences depending on whether B and B are of the same turn. We say that B follows (resp. covers) B