Madras, Neal N.Kazazi, Albi2026-03-102026-03-102025-09-242026-03-10https://hdl.handle.net/10315/43571An m×n grid graph is the induced subgraph of the square lattice whose vertex set consists of all integer grid points {(i, j) : 0 ≤ i < m, 0 ≤ j < n}. Let H and K be Hamiltonian cycles in an m × n grid graph G. We study the problem of reconfiguring H into K using a sequence of local transformations called moves. A box of G is a unit square face. A box with vertices a, b, c, d is switchable in H if exactly two of its edges belong to H, and these edges are parallel. Given such a box with edges ab and cd in H, a switch move removes ab and cd, and adds bc and ad. A double-switch move consists of performing two consecutive switch moves. If, after a double-switch move, we obtain a Hamiltonian cycle, we say that the double-switch move is valid. We prove that any Hamiltonian cycle H can be transformed into any other Hamiltonian cycle K via a sequence of valid double-switch moves, such that every intermediate graph remains a Hamiltonian cycle. This result extends to Hamiltonian paths. In that case, we also use single-switch moves and a third operation, the backbite move, which enables the relocation of the path endpoints.Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.MathematicsComputer scienceElectronic Thesis or Dissertation2026-03-10Hamiltonian pathsHamiltonian cyclesGrid graphsReconfiguration problemsSelf-avoiding walksGraph algorithmsMarkov chainsErgodicity