Abstract
A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices. In the past, we proved the Hamiltonian path and cycle problems for general supergrid graphs to be NP-complete. However, they are still open for solid supergrid graphs. In this paper, first we will verify the Hamiltonian cycle property of C-shaped supergrid graphs, which are a special case of solid supergrid graphs. Next, we show that C-shaped supergrid graphs are Hamiltonian connected except in a few conditions. For these excluding conditions of Hamiltonian connectivity, we compute their longest paths. Then, we design a linear-time algorithm to solve the longest path problem in these graphs. The Hamiltonian connectivity of C-shaped supergrid graphs can be applied to compute the optimal stitching trace of computer embroidery machines, and construct the minimum printing trace of 3D printers with a C-like component being printed.
Highlights
Supergrid Graphs in Linear Time.A Hamiltonian path in a graph is a spanning path of the graph
We focus on C-shaped supergrid graphs, which form a subclass of solid supergrid graphs
The Hamiltonian problems can be applied to many fields including computer science, biology, molecular science, etc
Summary
The Hamiltonian connectivity and longest path of shaped supergrid graphs can be applied for computing the optimal stitching trace of computer embroidery machines [12,16,17]. The Hamiltonian connectivity and longest path of these shaped letters play an important role in deciding these sewing traces They can be applied to construct the minimum printing trace of 3D printers. In [18], Itai et al proved that the Hamiltonian path problem on grid graphs is NP-complete They gave necessary and sufficient conditions for a rectangular grid graph having a Hamiltonian path between two given vertices. We will consider the Hamiltonian, Hamiltonian connectivity, and longest path of C-shaped supergrid graphs.
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