  A supergrid graph is a finite vertex-induced subgraph of the infinite graph
whose vertex set consists of all points of the plane with integer coordinates
and in which two vertices are adjacent if the difference of their x or y
coordinates is not larger than 1. The Hamiltonian path (cycle) problem is to
determine whether a graph contains a simple path (cycle) in which each vertex
of the graph appears exactly once. This problem is NP-complete for general
graphs and it is also NP-complete for general supergrid graphs. Despite the
many applications of the problem, it is still open for many classes, including
solid supergrid graphs and supergrid graphs with some holes. A graph is called
Hamiltonian connected if it contains a Hamiltonian path between any two
distinct vertices. In this paper, first we will study the Hamiltonian cycle
property of C-shaped supergrid graphs, which are a special case of rectangular
supergrid graphs with a rectangular hole. Next, we will show that C-shaped
supergrid graphs are Hamiltonian connected except few conditions. Finally, we
will compute a longest path between two distinct vertices in these graphs. The
Hamiltonian connectivity of C-shaped supergrid graphs can be applied to compute
the optimal stitching trace of computer embroidery machines, and construct the
minimum printing trace of 3D printers with a C-like component being printed.
