  Supergrid graphs contain grid graphs and triangular grid graphs as their
subgraphs. The Hamiltonian cycle and path problems for general supergrid graphs
were known to be NP-complete. A graph is called Hamiltonian if it contains a
Hamiltonian cycle, and is said to be Hamiltonian connected if there exists a
Hamiltonian path between any two distinct vertices in it. In this paper, we
first prove that every L-shaped supergrid graph always contains a Hamiltonian
cycle except one trivial condition. We then verify the Hamiltonian connectivity
of L-shaped supergrid graphs except few conditions. The Hamiltonicity and
Hamiltonian connectivity of L-shaped supergrid graphs can be applied to compute
the minimum trace of computerized embroidery machine and 3D printer when a
L-like object is printed. Finally, we present a linear-time algorithm to
compute the longest (s, t)-path of L-shaped supergrid graph given two distinct
vertices s and t.
