The Hamiltonian connectivity of some alphabet supergrid graphs

Supergrid graphs are first introduced by us and their structures are derived from grid and triangular-grid graphs. The Hamiltonian path problem on general supergrid graphs is a NP-complete problem. A graph is said to be Hamiltonian connected if a Hamiltonian path between any two nodes in it does exist. In the past, deciding whether or not a general supergrid graph contains a Hamiltonian path has been proved to be NP-complete. Very recently, we verified the Hamiltonian connectivity of some special supergrid graphs, including triangular, parallelogram, trapezoid, and rectangular supergrid graphs, except few conditions. In this paper, the Hamiltonian connectivity of alphabet supergrid graphs will be verifed. There are 26 types of alphabet supergrid graphs in which every capital letter is represented by a type of alphabet supergrid graphs. We will provide constructive proofs to verify the Hamiltonian connectivity of L-, F-, C-, and E-alphabet supergrid graphs. The results can be used to verify the Hamiltonian connectivity of other alphabet supergrid graphs with similar structure, such as G-, H-, J-, I-, O, P-, T-, S-, and U-alphabet supergrid graphs. The application of the Hamiltonian connectivity of alphabet supergrid graphs can be to compute the minimum stitching track of computer embroidery machines while a string is sewed into an object.