Treewidth
In graph theory, the treewidth of an undirected graph is a number associated with the graph. Treewidth may be defined in several equivalent ways: from the size of the largest vertex set in a tree decomposition of the graph, from the size of the largest clique in a chordal completion of the graph, from the maximum order of a haven describing a strategy for a pursuit-evasion game on the graph, or from the maximum order of a bramble, a collection of connected subgraphs that all touch each other.
Treewidth is commonly used as a parameter in the parameterized complexity analysis of graph algorithms. The graphs with treewidth at most k are also called partial k-trees; many other well-studied graph families also have bounded treewidth.
The concept of treewidth was originally introduced by Umberto Bertelé and Francesco Brioschi (1972) under the name of dimension. It was later rediscovered by Rudolf Halin (1976), based on properties that it shares with a different graph parameter, the Hadwiger number. Later it was again rediscovered by Neil Robertson and Paul Seymour (1984) and has since been studied by many other authors.
A tree decomposition of a graph G = (V, E) is a tree, T, with nodes X1, ..., Xn, where each Xi is a subset of V, satisfying the following properties (the term node is used to refer to a vertex of T to avoid confusion with vertices of G):
The width of a tree decomposition is the size of its largest set Xi minus one. The treewidth tw(G) of a graph G is the minimum width among all possible tree decompositions of G. In this definition, the size of the largest set is diminished by one in order to make the treewidth of a tree equal to one.
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