Concept

Tree decomposition

Summary
In graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the graph. Tree decompositions are also called junction trees, clique trees, or join trees. They play an important role in problems like probabilistic inference, constraint satisfaction, query optimization, and matrix decomposition. The concept of tree decomposition was originally introduced by . Later it was rediscovered by and has since been studied by many other authors. Definition Intuitively, a tree decomposition represents the vertices of a given graph G as subtrees of a tree, in such a way that vertices in G are adjacent only when the corresponding subtrees intersect. Thus, G forms a subgraph of the intersection graph of the subtrees. The full intersection graph is a chordal graph. Each subtree associates a graph vertex with a set of tree nodes.
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