Universal Sets of n Points for 1-bend Drawings of Planar Graphs with n Vertices ⋆ Hazel Everett 1 , Sylvain Lazard 1 , Giuseppe Liotta 2 , and Stephen Wismath 3 1 LORIA, INRIA Lorraine, Nancy Universit´ e, Nancy, France. {Hazel.Everett, Sylvain.Lazard}@loria.fr 2 Dip. di Ingegneria Elettronica e dell’Informazione, Universit`a degli Studi di Perugia liotta@diei.unipg.it 3 Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, Canada. wismath@cs.uleth.ca Abstract. This paper shows that any planar graph with n vertices can be point-set embedded with at most one bend per edge on a universal set of n points in the plane. An implication of this result is that any number of planar graphs admit a simultaneous embedding without mapping with at most one bend per edge. 1 Introduction Let S be a set of m distinct points in the plane and let G be a planar graph with n vertices (n  m). A point-set embedding of G on S is a planar drawing of G such that each vertex is drawn as a point of S and the edges are drawn as poly-lines. The problem of computing point-set embeddings of planar graphs has a long tradition both in the graph drawing and in the computational geometry literature (see, e.g., [5, 6, 8]). Considerable attention has been devoted to the study of universal sets of points. A set S of m points is said to be h-bend universal for the family of planar graphs with n vertices (n  m) if any graph in the family admits a point-set embedding onto S that has at most h bends along each edge. Gritzman, Mohar, Pach and Pollack [5] proved that every set of n distinct points in the plane is 0-bend universal for the all outerplanar graphs with n vertices. De Fraysseix, Pach, and Pollack [3] and independently Schnyder [9] proved that a grid with O(n 2 ) points is 0-bend universal for all planar graphs with n vertices. De Fraysseix et al. [3] also showed that a 0-bend universal set of points for all planar graphs having n vertices cannot have n + o( √ n) points. This last lower bound was improved by Chrobak and Karloff [2] and later by Kurowski [7] who showed that linearly many extra points are necessary for a 0- bend universal set of points for all planar graphs having n vertices. On the other hand, if two bends along each edge are allowed, a tight bound on the size of the ⋆ Research supported by NSERC and the MIUR Project “MAINSTREAM: Algo- rithms for Massive Information Structures and Data Streams”. Work initiated dur- ing the “Workshop on Graph Drawing and Computational Geometry”, Bertinoro, Italy, March 2007. We are grateful to the other participants, and in particular to W. Didimo and E. Di Giacomo, for useful discussions.