On Embedding a Graph on Two Sets of Points

Let S0, S1, . . . , Sk−1 be k sets of points such that the points of Si are colored with color i (i = 0, . . . , k−1). Let G be a planar graph such that |Si| vertices of G have color i, for every 0 ≤ i ≤ k − 1. A k-chromatic point-set embedding of G on S = S0 ∪ S1 ∪ · · · ∪ Sk−1 is a crossing-free drawing of G such that each vertex colored i is mapped to a point of Si, and each edge is a polygonal curve. If k = 1 (“monochromatic” case), all the vertices and all the points have the same color, and therefore any vertex can be mapped on any point. In this case the algorithm that computes the drawing of G on S can choose the mapping between vertices and points as it is more convenient. Kauffman and Wiese [4] prove that every planar graph admits a monochromatic point-set embedding with at most two bends per edge on any given set of points. If k = n, where n is the number of vertices of G, each vertex of G must be drawn on the unique point with the same color. Therefore the mapping between vertices and points is given as a part of the input and the drawing algorithm cannot change it. Pach and Wenger [5] prove that every planar graph has an n-chromatic point-set embedding on any given set of points such that each edge has O(n) bends; they also prove that this bound is asymptotically optimal in the worst case even if G is a path. Given the two results above, a natural question arises: If two bends per edge are necessary and sufficient to solve the monochromatic point-set embedding problem [4] while O(n) bends per edge are necessary and sufficient for the n-chromatic case [5], how many bends per edge do we need if the number of colors is a constant larger than one? In this paper we continue the study, initiated in a previous work [1], of the apparently simple case of two colors. The two colors will be referred in the following as red and blue and the two set of points S0 and S1 will be denoted as R and B. Notice that, by the result of Pach and Wenger [5], every planar graph has a bi-chromatic point-set embedding (2CPSE) with O(n) bends per edge (arbitrarily map every red/blue vertex to a red/blue point and then use the drawing technique of [5]). Therefore the question that we ask