Reconfiguring Simple s, t Hamiltonian Paths in Rectangular Grid Graphs

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? In this work, we introduce the notion of simple s, t Hamiltonian paths, and give an algorithm to reconfigure such paths of 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 in rectangular grid graphs.