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Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
A tree decomposition of a graph G is a tree T whose vertices are labelled by subsets of V(G) called bags, such that the two vertices of each edge are contained in at least one bag, and for each vertex v of G, the subgraph of T induced by the bags containing v is a connected subtree of T. The tree-width of a graph is one less than the minimum, over all tree decompositions, of the maximum bag size in that decomposition. The tree-width can be viewed as measuring how “tree-like” the graph is; the graphs of tree-width one are actually trees, while graphs of tree-width two are series-parallel graphs. The algorithm starts by heuristically finding a “good” tree decomposition for the graph (finding an optimal tree decomposition is NP-hard), and then using that tree decomposition to determine the order in which the vertices are processed, with the maximum bag size determining the maximum boundary size that will be encountered during this process. (Details of how the algorithm deals with gluing together the recursively calculated partial partition functions at a tree vertex of degree more than two are omitted here, but may be found in [85].)
A Novel Thresholding Approach to Find Out the Brain Tumor Region from MR Images
Published in IETE Journal of Research, 2021
On the off chance, Melekoodappattu et al. proposed that initially Fuzzy C Means algorithm has been utilized to point out the tumor region. Furthermore, the performance has been upgraded by extracting the features of the image by using the wavelet algorithm with the decomposition level of two and four. Also, during classification grasshopper optimization algorithm has been used to improve the performance [5]. Dollar et al. [6] proposed supervised learning calculation; because of an object and edge boundary recognition known as Boosted edge learning (BEL). The edge data is registered dependent on the pixel estimation of the image. Nikou et al. present a technique dependent on the hierarchical and spatially variant mixture model. Here the pixels are marked relying upon the random values and smoothness prior [7]. At the same time, the automatic threshold value calculation has been done by finding the histogram and quad-tree decomposition [8]. Another work stated that the Grab cut algorithm has variegated shortfalls that if the background has not been good enough, the poor result will yield. So for segmentation, the author [9] combines both grab cut and graph-based algorithm to identify the tumorous region. Another work stated by Wang et al. that; image color and texture regions have been registering in the directed weighted graphs. Then Mean shift clustering algorithm has utilized to improve the effectiveness of the image by comparing the pairwise region [10].
A novel numerical method for steady-state thermal simulation based on loop-tree and HBRWG basis functions
Published in Numerical Heat Transfer, Part B: Fundamentals, 2020
Liang Chen, Min Tang, Zuhui Ma, Junfa Mao
Many analytical and numerical methods [7–14] have been proposed to solve 2-D heat conduction with different types of boundary conditions, including temperature, heat flux and convective boundary conditions. Although the analytical techniques have the advantage of high efficiency and accuracy, these closed-form expressions are available only for special geometries with homogeneous material. The numerical methods, such as the finite-volume method (FVM), finite-difference method (FDM), and finite-element method (FEM), are more commonly used to carry out accurate thermal simulation for complex structures of electronic system even though they may be time consuming. In recent years, a fast numerical approach for solving the Poisson’s equation has been presented to analyze the electrostatic problems in [15–17]. The computational efficiency can be improved significantly using the loop-tree decomposition technique. However, this approach is only suitable for addressing the problems with the Dirichlet or zero Neumann boundary conditions, which restricts its application for solving steady-state heat conduction equation with arbitrary boundaries.
Product platforms design, selection and customisation in high-variety manufacturing
Published in International Journal of Production Research, 2020
Francesco Gabriele Galizia, Hoda ElMaraghy, Marco Bortolini, Cristina Mora
The third step of the proposed decision support system (DSS) manages the phylogenetic tree decomposition supporting the product platforms selection process. Product platforms have to be designed and selected to maximise the number of components in each platform in order to reduce the number of assembly/disassembly tasks to be performed to obtain the desired product variant while minimising the number of different platforms to be assembled and stored in order to reduce variety, inventory costs and storage space. Step III addresses this trade-off: the phylogenetic tree obtained in the second step (Figure 1) is decomposed into multiple levels (Figure 3) from the native platforms (Level 1) to the final variants (Level L). A native platform is a platform that has no incoming arrows (PL1 in the referenced example), while a platform or a product variant belongs to level L if it does not have outgoing arrows (from P6 to P10 in the referenced example).