Recently, we have introduced two graph-decomposition theorems based on a new graph product, motivated by applications in the context of synchronising periodic real-time processes. This vertexremoving synchronised product (VRSP) is based on modifications of the well-known Cartesian product and is closely related to the synchronised product due to Wohrle and Thomas. Here, we ¨ show how we can relax the requirements of these two graph-decomposition theorems.
DOCUMENT
Recently, we have introduced and modified two graph-decomposition theorems based on a new graph product, motivated by applications in the context of synchronising periodic real-time processes. This vertex-removing synchronised product (VRSP), is based on modifications of the well-known Cartesian product and is closely related to the synchronised product due to Wohrle and Thomas. Here, we recall the definition of the VRSP and the two modified graph-decompositions and introduce three new graph-decomposition theorems. The first new theorem decomposes a graph with respect to the semicomplete bipartite subgraphs of the graph. For the second new theorem, we introduce a matrix graph, which is used to decompose a graph in a manner similar to the decomposition of graphs using the Cartesian product. In the third new theorem, we combine these two types of decomposition. Ultimately, the goal of these graph-decomposition theorems is to come to a prime-graph decomposition.
MULTIFILE
Recently, we have introduced two graph-decomposition theorems based on a new graph product, motivated by applications in the context of synchronising periodic real-time processes. This vertex-removing synchronised product (VRSP) is based on modifications of the well-known Cartesian product and is closely related to the synchronised product due to Wöhrle and Thomas. Here, we recall the definition of the VRSP and the two graph-decomposition theorems, we relax the requirements of these two graph-decomposition theorems and prove these two (relaxed) graph-decomposition theorems.
DOCUMENT
Many lithographically created optical components, such as photonic crystals, require the creation of periodically repeated structures [1]. The optical properties depend critically on the consistency of the shape and periodicity of the repeated structure. At the same time, the structure and its period may be similar to, or substantially below that of the optical diffraction limit, making inspection with optical microscopy difficult. Inspection tools must be able to scan an entire wafer (300 mm diameter), and identify wafers that fail to meet specifications rapidly. However, high resolution, and high throughput are often difficult to achieve simultaneously, and a compromise must be made. TeraNova is developing an optical inspection tool that can rapidly image features on wafers. Their product relies on (a) knowledge of what the features should be, and (b) a detailed and accurate model of light diffraction from the wafer surface. This combination allows deviations from features to be identified by modifying the model of the surface features until the calculated diffraction pattern matches the observed pattern. This form of microscopy—known as Fourier microscopy—has the potential to be very rapid and highly accurate. However, the solver, which calculates the wafer features from the diffraction pattern, must be very rapid and precise. To achieve this, a hardware solver will be implemented. The hardware solver must be combined with mechatronic tracking of the absolute wafer position, requiring the automatic identification of fiduciary markers. Finally, the problem of computer obsolescence in instrumentation (resulting in security weaknesses) will also be addressed by combining the digital hardware and software into a system-on-a-chip (SoC) to provide a powerful, yet secure operating environment for the microscope software.
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