On local transformations in plane geometric graphs embedded on small grids

Abstract

Given two n-vertex plane graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ with $|E_1|=|E_2|$ embedded in the $n\times n$ grid, with straight-line segments as edges, we show that with a sequence of $\mathrm{O}(n)$ point moves (all point moves stay within a $5n\times 5n$ grid) and $\mathrm{O}(n^2)$ edge moves, we can transform the embedding of $G_1$ into the embedding of $G_2$. In the case of n-vertex trees, we can perform the transformation with $\mathrm{O}(n)$ point and edge moves with all moves staying in the $n\times n$ grid. We prove that this is optimal in the worst case as there exist pairs of trees that require $\mathrm{\Omega }(n)$ point and edge moves. We also study the equivalent problems in the labeled setting.