The Longest $(s, t)$-paths of $O$-shaped Supergrid Graphs

In this paper, we continue the study of the Hamiltonian and longest $(s, t)$-paths of supergrid graphs. The Hamiltonian $(s, t)$-path of a graph is a Hamiltonian path between any two given vertices $s$ and $t$ in the graph, and the longest $(s, t)$-path is a simple path with the maximum number of vertices from $s$ to $t$ in the graph. A graph holds Hamiltonian connected property if it contains a Hamiltonian $(s, t)$-path. These two problems are well-known NP-complete for general supergrid graphs. An $O$-shaped supergrid graph is a special kind of a rectangular grid graph with a rectangular hole. In this paper, we first prove the Hamiltonian connectivity of $O$-shaped supergrid graphs except few conditions. We then show that the longest $(s, t)$-path of an $O$-shaped supergrid graph can be computed in linear time. The Hamiltonian and longest $(s, t)$-paths of $O$-shaped supergrid graphs can be applied to compute the minimum trace of computerized embroidery machine and 3D printer when a hollow object is printed.