Let be a planar graph with vertices and with a partition of the vertex set into subsets for some positive integer . Let be a set of distinct points in the plane with a partition into subsets with (). This paper studies the problem of computing a planar polyline drawing of , such that each vertex of is mapped to a distinct point of . Lower and upper bounds on the number of bends per edge are proved for any . In the special case , we improve the upper and lower bounds presented in a paper by Pach and Wenger [J. Pach, R. Wenger, Embedding planar graphs at fixed vertex locations, Graphs and Combinatorics 17 (2001) 717–728]. The upper bound is based on an algorithm for computing a topological book embedding of a planar graph, such that the vertices follow a given left-to-right order and the number of crossings between every edge and the spine is asymptotically optimal, which can be regarded as a result of independent interest.
An extended abstract of this paper appeared in the proceedings of the 10th Workshop on Algorithms and Data Structures, WADS 2007. Work partially supported by the MIUR Project “MAINSTREAM: Algorithms for massive information structures and data streams”.