As a measure for the resemblance of curves in arbitrary dimensions we consider the so-called Fréchet-distance, which is compatible with parametrizations of the curves. For polygonal chains P and Q consisting of p and q edges an algorithm of runtime O(pq log(pq)) measuring the Fréchet-distance between P and Q is developed. Then some important variants are considered, namely the Fréchet-distance for closed curves, the nonmonotone Fréchet-distance and a distance function derived from the Fréchet-distance measuring whether P resembles some part of the curve Q.

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Abstract. This paper studies 3-dimensional visibility representations of graphs in which objects in 3-d correspond to vertices and vertical visibilities between these objects correspond to edges. We ask which classes of simple objects are universal, i.e. powerful enough to represent all graphs.In particular, we show that there is no constant k for which the class of all polygons having k or fewer sides is universal. However, we show by construction that every graph on n vertices can be represented by polygons each having at most 2n sides. The construction can be carried out by an O(n 2) algorithm. We also study the universality of classes of simple objects (translates of a single, not necessarily polygonal object) relative to cliques Kn and similarly relative to complete bipartite graphs K,~,m.

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In this paper it will be shown that the following problem is NP-hard. We are given a labeled planar graph, each vertex of which is assigned to a disc in the plane. Decide whether it is possible to embed the graph in the plane with line segments as edges such that each vertex lies in its disc.

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