On local transformations in plane geometric graphs embedded on small grids

Given two -vertex plane graphs and with embedded in the grid, with straight-line segments as edges, we show that with a sequence of point moves (all point moves stay within a grid) and edge moves, we can transform the embedding of into the embedding of . In the case of -vertex trees, we can perform the transformation with point and edge moves with all moves staying in the grid. We prove that this is optimal in the worst case as there exist pairs of trees that require point and edge moves. We also study the equivalent problems in the labeled setting.