We study the following reconfiguration problem: given two $s,t$ Hamiltonian paths connecting diagonally opposite corners $s$ and $t$ of a rectangular grid graph $G$, can we transform one to the other using only {local} operations in the grid cells? Here we introduce the notion of simple $s,t$ Hamiltonian paths, and give an algorithm to reconfigure such paths in $G$ in $O(|G|)$ time using local operations in unit grid cells. We achieve our algorithmic result by proving a combinatorial structure theorem for simple $s,t$ Hamiltonian paths.

https://link.springer.com/chapter/10.1007/978-3-030-79987-8_35