Drawing colored graphs on colored points

Abstract

Let G be a planar graph with n vertices and with a partition of the vertex set into subsets V 0 , … , V k − 1 for some positive integer 1 ≤ k ≤ n . Let S be a set of n distinct points in the plane with a partition into subsets S 0 , … , S k − 1 with ∣ V i ∣ = ∣ S i ∣ ( 0 ≤ i ≤ k − 1 ). This paper studies the problem of computing a planar polyline drawing of G , such that each vertex of V i is mapped to a distinct point of S i . Lower and upper bounds on the number of bends per edge are proved for any 2 ≤ k ≤ n . In the special case k = n , 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.