Let S be a point set in general position on the plane such that its elements are colored red or blue. We study the following problem: Remove as few points as possible from S such that the remaining points can be enclosed by two isothetic rectangles, one containing all the red points, the other all the blue points, and such that each rectangle contains only points of one color. We prove that this problem can be solved in O(n 2 log n) time and O(n) space. We show how our techniques can be generalized to solve other variants of the given problem such as the 3-dimensional problem and the trichromatic problem.
We show that any two outer-triangulations on the same closed surface can be transformed into each other by a sequence of diagonal ips, up to isotopy, if they have a su ciently large and equal number of vertices.
Abstract.A graph G is said to be grid locatable if it admits a representation such that vertices are mapped to grid points and edges to line segments that avoid grid points but the extremes. Additionally G is said to be properly embeddable in the grid if it is grid locatable and the segments representing edges do not cross each other. We study the area needed to obtain those representations for some graph families.
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