| Research Abstract |
We mainly research the following two topics, and also study some related topics in graph theory.
For given graph G and set S of points in the plane, we want to embed G onto S so that each edge of G is a straight line segment, and each vertex is a point in S. If possible, we want to find such a embedding without crossings, if it is impossible to embed G onto S without crossings, we want to find a embedding with a small number of crossings.
We obtained some results on this topics by using graph theorict method and balanced partition methods. There are still some interesting unsolved problems, our results developed this area.
Another research topic is balanced partition problems. Namely, give a set of red points and a set of blue points in the plane, we want to divide the plane into k disjoint convex subsets so that each subset contains n_1 red points and n_i×m blue points under the assumption that n_1 + ・・・n_k red points and m(n_1 + ・・・ + n_k) blue points are given. If n_1 =・・・ =n_k, then this problem was partially solved by us and complete solved by three groups of researchers. We obtained some more general and related results on this problem.
The above two problems have a relation ship, and we wrote a survey entitled "Discrete Geometry on Red and Blue Points in the Plane - A Survey", which includes the above two topics as main parts. So these research area are new research area but becomes popular very fast.
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