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.

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In this paper, we give the necessary and sufficient conditions for the existence of Hamiltonian paths in L−alphabet and C−alphabet grid graphs. We also present a linear-time algorithm for finding Hamiltonian paths in these graphs.

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In this paper, first we give a sequential linear-time algorithm for the longest path problem in meshes. This algorithm can be considered as an improvement of [13]. Then based on this sequential algorithm, we present a constanttime parallel algorithm for the problem which can be run on every parallel machine.