Finding Hamiltonian and Longest (s,t)-Paths of C-Shaped Supergrid Graphs in Linear Time

A graph is called Hamiltonian connected if it contains a path between any two distinct vertices. In the past, we proved and cycle problems for general supergrid graphs to be NP-complete. However, they are still open solid graphs. this paper, first will verify property of C-shaped graphs, which special case Next, show that except in few conditions. For these excluding conditions connectivity, compute their longest paths. Then, design linear-time algorithm solve problem The connectivity can applied optimal stitching trace computer embroidery machines, construct minimum printing 3D printers with C-like component being printed.