Suppose that we are given a forest F of k rooted trees T_1,...,T_k, a set of n points in the plane that are in general position, and an ordered subset p_1,...,p_k of these points.
Problem: Can we find a geometric realization of F with no crossing edges such that the root of T_i is at point p_i.
Kaneko and Kano [KK1,KK2,KK3] have found such realizations in some special cases, including a) k = 2, b) each tree is a star, and c) all of the trees are of the same size. However, they conjecture that the problem is false in general.
[KK1] A. Kaneko and M. Kano, Straight-line embeddings of two rooted trees in the plane, preprint.
[KK2] A. Kaneko and M. Kano, Straight-line embeddings of rooted star forests in the plane, preprint.
[KK1] A. Kaneko and M. Kano, A balanced partition of points in the plane
and tree embedding problems, preprint.
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