{Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewise-differentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular, this result settles the well-studied carpenter’s rule conjecture.

Every three-connected planar graph with n vertices has a drawing on an O(n^2) \times O(n^2) grid in which all faces are strictly convex polygons. These drawings are obtained by perturbing (not strictly) convex drawings on O(n) \times O(n) grids. Tighter bounds are obtained when the faces have fewer sides. In the proof, we derive an explicit lower bound on the number of primitive vectors in a triangle.

Given a set of nonintersecting polygonal obstacles in the plane, thelink distance between two pointss andt is the minimum number of edges required to form a polygonal path connectings tot that avoids all obstacles. We present an algorithm that computes the link distance (and a corresponding minimum-link path) between two points in timeO(Eα(n) log² n) (and spaceO(E)), wheren is the total number of edges of the obstacles,E is the size of the visibility graph, and α(n) denotes the extremely slowly growing inverse of Ackermann's function. We show how to extend our method to allow computation of a tree (rooted ats) of minimum-link paths froms to all obstacle vertices. This leads to a method of solving the query version of our problem (for query pointst).

We improve the lower bounds on Klarner's constant, which describes the exponential growth rate of the number of polyominoes (connected subsets of grid squares) with a given number of squares. We achieve this by analyzing polyominoes on a different surface, a so-called $\textit{twisted cylinder}$ by the transfer matrix method. A bijective representation of the "states'' of partial solutions is crucial for allowing a compact representation of the successive iteration vectors for the transfer matrix method.