The hamiltonian path graph is connected for simple s, t paths in rectangular grid graphs

An s, t Hamiltonian path P for an \(m \times n\) rectangular grid graph \(\mathbb {G}\) is a Hamiltonian path from the top-left corner s to the bottom-right corner t. We define an operation “square-switch” on s, t Hamiltonian paths P affecting only those edges of P that lie in some small (2 units by 2 units) square subgrid of \(\mathbb {G}\). We prove that when applied to suitable locations, the result of the square-switch is another s, t Hamiltonian path. Then we use square-switch to achieve a reconfiguration result for a subfamily of s, t Hamiltonian paths we call simple paths, that has the minimum number of bends for each maximal internal subpath connecting any two vertices on the boundary of the grid graph. We give an algorithmic proof that the Hamiltonian path graph \(\mathcal {G}\) whose vertices represent simple paths is connected when edges arise from the square-switch operation: our algorithm reconfigures any given initial simple path P to any given target simple path \(P'\) in \(\mathcal {O}\)(\( |P |\)) time and \(\mathcal {O}\)(\( |P |\)) space using at most \({5} |P |/ {4}\) square-switches, where \( |P |= m \times n\) is the number of vertices in the grid graph \(\mathbb {G}\) and hence in any Hamiltonian path P for \(\mathbb {G}\). Thus the diameter of the simple path graph \(\mathcal {G}\) is at most 5mn/ 4 for the square-switch operation, which we show is asymptotically tight for this operation.