Bipartite Embeddings of Trees in the Plane

Given a tree T on n vertices and a set P of n points in the plane in general position, it is known that T can be straight line embedded in P without crossings. The problem becomes more diicult if T is rooted and we want to root it at any particular point of P. The problem in this form was posed by Perles and partially solved by Pach and Torosick 5]. A complete solution was found by Ikebe et al. 4]. A related result by A. Tamura and Y. Tamura 6] is that, given a point set P = fp 1 ; : : : ; p n g and a sequence d = (d 1 ; : : : d n) of positive integers with P d i = 2n ? 2, there exists an embedding of some tree in P such that the degree of p i is equal to d i. Optimal algorithms for solving the above problems have been found by Bose, McAllister and Snoeyink 1]. In this paper we consider the following embedding problem. A point set P in the plane in general position (no three points collinear) is partitioned into two disjoint sets R and B (the red and the blue points), and we are asked to embed a tree T in P without crossings and with the additional property that all the edges are red/blue, i.e. all edges connect a point in R to another point in B. We call such an embedding a bipartite embedding of T with respect to the bipartition (R; B). Any tree T is a bipartite graph and the bipartition V (T) = (V 1 ; V 2) induced in the vertex set is in fact unique. An obvious necessary condition for the existence of a bipartite embedding of T in P is that the cardinalities of both bipartitions, those of T and of P , match correctly. However, simple examples shows that this is not always suucient. It is then natural to relax the requirements of the problem and to ask: given a bipartition (R; B), is it always possible to nd a bipartite embedding of some tree with respect to (R; B)? It is straightforward to prove that the answer is aarmative. Take any red point and join it to all the blue points. It is then clear that the remaining red points can be …