Tree decomposition

This article is about tree structure of graphs. For decomposition of graphs into trees, see Graph theory § Decomposition problems. For decomposition of trees in nature, see Nurse log.
A graph with eight vertices, and a tree decomposition of it onto a tree with six nodes. Each graph edge connects two vertices that are listed together at some tree node, and each graph vertex is listed at the nodes of a contiguous subtree of the tree. Each tree node lists at most three vertices, so the width of this decomposition is two.

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,[1] and matrix decomposition.

The concept of tree decomposition was originally introduced by Rudolf Halin (1976). Later it was rediscovered by Neil Robertson and Paul Seymour (1984) and has since been studied by many other authors.[2]

  1. ^ Gottlob et al. (2012).
  2. ^ Diestel (2005) pp.354–355