Drawing colored graphs on colored points

Abstract

Let G be a planar graph with n vertices whose vertex set is partitioned into subsets V0,..., Vk-1 for a positive integer 1 ≤ k ≤ n and let S be a set of n distinct points in the plane partitioned into subsets S0...,Sk-1 with |Vi| = |Si| 0 ≤ i ≤ k - 1). This paper studies the problem of computing a crossing-free drawing of G such that each vertex of Vi is mapped to a distinct point of Si. Lower and upper bounds on the number of bends per edge are proved for any 3 ≤ k ≤ n. As a special case, we improve the upper and lower bounds presented in a paper by Pach and Wenger for k = n [Graphs and Combinatorics (2001), 17:717-728]. © Springer-Verlag Berlin Heidelberg 2007.