On Local Transformations in Plane Geometric Graphs Embedded on Small Grids

Given two n-vertex plane graphs G 1 and G 2 embedded in the n × n grid with straight-line segments as edges, we show that with a sequence of O(n) point moves (all point moves stay within a 5n× 5n grid) and O(n 2) edge moves, we can transform G 1 into G 2. In the case of n-vertex trees, we can perform the transformation with O(n) point and edge moves, and show this is optimal. We also study the equivalent problems in the labelled setting.