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

The Hamiltonicity, Hamiltonian Connectivity, and Longest (s, t)-path of L-shaped Supergrid Graphs