The Longest (s, t)-Path Problem on O-Shaped Supergrid Graphs

The Longest (s, t)-Path Problem on O-Shaped Supergrid Graphs

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The longest (s,t)-path problem on supergrid graphs is known to be NP-complete.However, the complexity of this problem on supergrid graphs with or without holes is still unknown.In the past, we presented linear-time algorithms for solving the longest (s,t)-path 3 gallon flat back bucket problem on L-shaped and C-shaped supergrid graphs, which form subclasses of supergrid graphs without holes.In this paper, we will determine the complexity of the longest (s,t)-path problem on O-shaped supergrid graphs, which form a subclass of supergrid graphs with holes.

These graphs are rectangular supergrid graphs with rectangular holes.It is worth noting that O-shaped supergrid graphs contain L-shaped and C-shaped supergrid graphs as subgraphs, but there is no inclusion relationship between them.We will propose a linear-time algorithm to solve the longest (s,t)-path problem on O-shaped supergrid graphs.The longest (s,t)-paths of O-shaped supergrid graphs have applications in calculating the minimum trace when sweet nothings acai bowl printing hollow objects using computer embroidery machines and 3D printers.