Geodesic
1-Complex $s,t$ Hamiltonian Paths: Structure and Reconfiguration in Rectangular Grids
Journal of Graph Algorithms and Applications, Special Issue on Selected Papers from the 16th International Conference and Workshops on Algorithms and Computation, WALCOM 2022 , Tome 27 (2023) no. 4, pp. 281-327.

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We give a complete structure theorem for $1$-complex $s,t$ Hamiltonian paths in rectangular grid graphs. We use the structure theorem to design an algorithm to reconfigure one such path into any other in linear time, making a linear number of switch operations in grid cells. 
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     author = {Rahnuma Islam Nishat and Venkatesh Srinivasan and Sue Whitesides},
     title = {1-Complex $s,t$ {Hamiltonian} {Paths:}  {Structure} and {Reconfiguration} in {Rectangular} {Grids}},
     journal = {Journal of Graph Algorithms and Applications},
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Rahnuma Islam Nishat; Venkatesh Srinivasan; Sue Whitesides. 1-Complex $s,t$ Hamiltonian Paths:  Structure and Reconfiguration in Rectangular Grids. Journal of Graph Algorithms and Applications, 
							Special Issue on Selected Papers from the 16th International Conference and  Workshops on Algorithms and Computation, WALCOM 2022
					, Tome 27 (2023) no. 4, pp. 281-327. doi : 10.7155/jgaa.00624. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00624/

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