We prove that every tree T = (V, E) on n vertices with edges of unit length can be embedded in the plane with distortion O( √ n); that is, we construct a mapping f : where ρ(u, v) denotes the length of the path from u to v in T . The embedding is described by a simple and easily computable formula. This is asymptotically optimal in the worst case. We also construct interesting optimal embeddings for a special class of trees (fans consisting of paths of the same length glued together at a common vertex).Communicated by: P. Mutzel and M. Jünger;

In this paper we study 3-dimensional visibility representations of complete graphs. The vertices are represented by equal regular polygons lying in planes parallel to the xy-plane. Two vertices are adjacent if and only if the two corresponding polygons see each otheri.e. it is possible to construct an abscissa perpendicular to the xy-plane connecting the two polygons and avoiding all the others. We give the bounds for the maximal size f (k) of a clique represented by regular k-gons: k+1 2 + 2 ≤ f (k) ≤ 2 2 k and we present a particular result for triangles: f (3) ≥ 14.

No abstract

We prove that every tree T = (V, E) on n vertices with edges of unit length can be embedded in the plane with distortion O( √ n); that is, we construct a mapping f : where ρ(u, v) denotes the length of the path from u to v in T . The embedding is described by a simple and easily computable formula. This is asymptotically optimal in the worst case. We also construct interesting optimal embeddings for a special class of trees (fans consisting of paths of the same length glued together at a common vertex).Communicated by: P. Mutzel and M. Jünger;

Abstract.We prove that every tree T = (V, E) on n vertices can be embedded in the plane with distortion O( √ n); that is, we construct a where ρ(u, v) denotes the length of the path from u to v in T (the edges have unit lengths). The embedding is described by a simple and easily computable formula. This is asymptotically optimal in the worst case. We also prove several related results.