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In this paper, we first introduce a novel class of graphs, namely supergrid. Supergrid graphs include grid graphs and triangular grid graphs as their subgraphs. The Hamiltonian cycle and path problems for grid graphs and triangular grid graphs were known to be NP-complete. However, they are unknown for supergrid graphs. The Hamiltonian cycle (path) problem on supergrid graphs can be applied to control the stitching traces of computerized sewing machines. In this paper, we will prove that the Hamiltonian cycle problem for supergrid graphs is NP-complete. It is easily derived from the Hamiltonian cycle result that the Hamiltonian path problem on supergrid graphs is also NP-complete. We then show that two subclasses of supergrid graphs, including rectangular (parallelism) and alphabet, always contain Hamiltonian cycles.

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A Hamiltonian path of a graph G is a simple path that contains each vertex of G exactly once. A Hamiltonian cycle of a graph is a simple cycle with the same property. The Hamiltonian path (resp. cycle) problem involves testing whether a Hamiltonian path (resp. cycle) exists in a graph. The 1HP (resp. 2HP) problem is to determine whether a graph has a Hamiltonian path starting from a specified vertex (resp. starting from a specified vertex and ending at the other specified vertex). The Hamiltonian problems include the Hamiltonian path, Hamiltonian cycle, 1HP, and 2HP problems. A graph is a distance-hereditary graph if each pair of vertices is equidistant in every connected induced subgraph containing them. In this paper, we present a unified approach to solving the Hamiltonian problems on distance-hereditary graphs in linear time.

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.