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The exact complexity of the Tutte polynomial
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Tomer Kotek, Johann A. Makowsky
Tree-width and clique-width are graph parameters which measure the compositionality of a graph. Tree-width is usually defined as the minimum width of a certain map of a graph into a tree called a tree-decomposition (see, e.g., [383]). In contrast, graphs of clique-width k are defined inductively. For uniformity of presentation, we also give an inductive definition of graphs of tree-width k.
Decentralized Data Fusion
Published in David L. Hall, Chee-Yee Chong, James Llinas, Martin Liggins, Distributed Data Fusion for Network-Centric Operations, 2013
Paul Thompson, Eric Nettleton, Hugh Durrant-Whyte
A complete k-tree graph is made up of cliques of k + 1 nodes [13,14]. Each adjacent pair of cliques overlaps at k nodes (a junction or separator). The overall graph of connections between the cliques is a tree. A k-tree graph has treewidth of k, so called because the separators are made of k nodes.
Modelling the spatial distribution of heavy vehicle loads on long-span bridges based on undirected graphical model
Published in Structure and Infrastructure Engineering, 2019
Zhicheng Chen, Yuequan Bao, Jiahui Chen, Hui Li
When the number of nodes in the vehicle-location UGM is large, the joint PMF cannot be directly calculated via Equation (4) because it involves the calculation of the partition function which requires specifying all possible combinations of realizations of the random vector given in Equation (2). An alternative approach is using the junction tree algorithm, which can provide an exact inference procedure for arbitrary graphical models and highly efficient if the treewidth is small (Murphy, 2012). The so-called treewidth is defined as the number of nodes contained in the largest clique of the junction tree. The junction tree built from the vehicle-location UGM has a small treewidth being equal to where is the number of lanes.