Consider the problem of moving a closed chain of n links in two or more dimensions from one given configuration to another. The links have fixed lengths and may rotate about their endpoints, possibly passing through one another. The notion of a "line-tracking motion" is defined, and it is shown that when reconfiguration is possible by any means, it can be achieved by O(n) line-tracking motions. These motions can be computed in O(n) time on real RAM. It is shown that in three or more dimensions, reconfiguration is always possible, but that in dimension two this is not the case. Reconfiguration is shown to be always possible in two dimensions if and only if the sum of the lengths of the second and third longest links add to at most the sum of the lengths of the remaining links. An O(n) algorithm is given for determining whether it is possible to move between two given configurations of a closed chain in the plane and, if it is possible, for computing a sequence of line-tracking motions to carry out the reconfiguration. * The research of the second author was partially supported by an NSERC operating grant.

Abstract.We consider contact representations of graphs where vertices are represented by cuboids, i.e. interior-disjoint axis-aligned boxes in 3D space. Edges are represented by a proper contact between the cuboids representing their endvertices. Two cuboids make a proper contact if they intersect and their intersection is a non-zero area rectangle contained in the boundary of both. We study representations where all cuboids are unit cubes, where they are cubes of different sizes, and where they are axis-aligned 3D boxes. We prove that it is NP-complete to decide whether a graph admits a proper contact representation by unit cubes. We also describe algorithms that compute proper contact representations of varying size cubes for relevant graph families. Finally, we give two new simple proofs of a theorem by Thomassen stating that all planar graphs have a proper contact representation by touching cuboids.

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Abstract. Increasing attention has been given recently to drawings of graphs in which edges connect vertices based on some notion of proximity. Among such drawings are Gabriel, relative neighborhood, Delaunay, sphere of influence, and minimum spanning drawings. This paper attempts to survey the work that has been done to date on proximity drawings, along with some of the problems which remain open in this area. Proximity DrawingsIn 1969, Gabriel and Sokal [15] presented a method for associating a graph to a set of geographic data points P by connecting points z, y E P with an edge if and only if the closed disk having the segment ~ as diameter contained no other point of P. This graph, now called the Gabriel graph of P, is just one example of what have come to be called proximity graphs. Loosely speaking, a proximity graph is a graph constructed from a set P of points in some metric space by connecting pairs of points which are deemed to be "sufficiently" close together. A set P can give rise to a variety of different proximity graphs depending upon the definition of closeness used. Early work in this area was concerned for the most part with the problems of determining notions of proximity which might best capture the "internal structure" of a set of points and, having done so, of efficiently computing the proximity graph of a given set of points. For a survey of such results, see Jaromczyk and Toussaint [21].More recently, increasing attention has been given to the proximity drawing problem: given a graph G and a definition of proximity, determine whether a set P of points exists such that the proximity graph of P is the given graph, and if so, compute such a set. Clearly the set P, if it exists, gives rise to a straight-line

The link center of a simple polygon P is the set of points x inside P at which the maximal link-distance from x to any other point in P is minimized. Here the link distance between two points x, y inside P is defined to be the smallest number of straight edges in a polygonal path inside P connecting x to y. We prove several geometric properties of the link center and present an algorithm that calculates this set in time O(n2), where n is the number of sides of P. We also give an