On unfolding lattice polygons/trees and diameter-4 trees

Abstract

We consider the problems of straightening polygonal trees and convexifying polygons by continuous motions such that rigid edges can rotate around vertex joints and no edge crossings are allowed. A tree can be straightened if all its edges can be aligned along a common straight line such that each edge points "away" from a designated leaf node. A polygon can be convexified if it can be reconfigured to a convex polygon. A lattice tree (resp. polygon) is a tree (resp. polygon) containing only edges from a square or cubic lattice. We first show that a 2D lattice chain or a 3D lattice tree can be straightened efficiently in O(n) moves and time, where n is the number of tree edges. We then show that a 2D lattice tree can be straightened efficiently in O(n2) moves and time. Furthermore, we prove that a 2D lattice polygon or a 3D lattice polygon with simple shadow can be convexified efficiently in O(n2) moves and time. Finally, we show that two special classes of diameter-4 trees in two dimensions can always be straightened. © Springer-Verlag Berlin Heidelberg 2006.