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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 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 printing hollow objects using computer embroidery machines and 3D printers.

The Hamiltonian path problem on general graphs is well-known to be NP-complete. In the past, we have proved it to be also NP-complete for supergrid graphs. A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices in it. Determining whether a supergrid graph is Hamiltonian connected is clear to be NPcomplete. Recently, we proved the Hamiltonian connectivity of some special supergrid graphs, including rectangular, triangular, parallelogram, and trapezoid. In this paper, we will study the Hamiltonian connectivity of alphabet supergrid graphs. There are 26 types of alphabet supergrid graphs in which every capital letter is represented by a type of alphabet supergrid graphs. We will prove L-, C-, F-, E-, N-, and F-alphabet supergrid graphs to be Hamiltonian connected. The Hamiltonian connectivity of the other alphabet supergrid graphs can be verified similarly. The Hamiltonian connected property of alphabet supergrid graphs can be applied to compute the minimum stitching trace of computerized embroidery machines during the sewing process.

Embroidery through computer aided semi-automatic machines is one of the most widely used option for the surface ornamentation of apparel fabrics at present. Since the embroidery process includes addition of certain amount of embroidery-threads depending upon the design motif, it is quite obvious that basic physical and functional properties of fabric are subject to change. It is therefore important to develop an algorithm or empirical equation for proper prediction of the properties of the embroidered fabric, relevant to its required end-use in apparel industry. In this context, an effort has been made to determine a prediction equation through linear multiple regressions for the prediction of longitudinal stiffness of embroidered fabric in terms of flexural rigidity in warp direction of the base fabric, considering the input parameters as warp-way flexural rigidity of the base fabric, breaking load and linear density of the embroidery thread, stitch density, average stitch length and average stitch angle of the embroidery design. The final Prediction model is statistically verified taking new embroidery samples of different varieties. It is found that the model can predict with a very satisfactory level of accuracy. Also, the influences of the embroidery parameters in this context have been analyzed through the corresponding regression coefficients and the three dimensional (3D) surface curves. Stitch density has been emerged as the most influential parameter, followed by the stitch length and the stitch angle.

Purpose>Embroidery as a textile embellishment technique plays an important role in people's daily life. Esthetic embroidery artworks possess cultural values. With the development of robotics and artificial intelligence (AI), these technologies have been studied and applied in the embroidery process. This study aims to survey how these technologies facilitate embroidery from different aspects.Design/methodology/approach>This paper surveys how the technologies of robotics and AI are applied in the embroidery field. The applications are mainly reviewed from three aspects: computerized robotic embroidery systems has been widely used for the mass production of embroidered textiles, the advanced technological systems and techniques have greatly facilitated the development of smart textiles and the artificial intelligence plays an important role in the inheritance, innovation and protection of traditional handicraft artwork of embroidery.Findings>The programmable robotic embroidery machines have greatly improved the production efficiency of embroidered textiles and promoted the development of electronic textiles. The AI, mainly the deep learning technology, brings significant benefits to esthetic embroidery creation. Technology-based embroidery has become a hot research topic in the field of textiles.Originality/value>This paper summarizes the application of robotics and AI technologies in the field of embroidery, which provides readers a comprehensive and systematic understanding about the research progress of modern technology-oriented embroidery. This helps readers gain inspiration from the technology perspectives.