The treewidth is a measure of the count of original graph vertices mapped onto any tree vertex in an optimal tree decomposition.
Determining the treewidth of an arbitrary graph is an NP-hard
problem. However, many NP-hard problems on graphs of bounded treewidth can be
solved in polynomial time.
An empty graph has treewidth 0, a tree or forest has treewidth 1, and graphs with treewidth at
most 2 correspond to series-parallel graphs. Every Halin
graph has a treewidth of 3 (Bodlaender 1988).
The treewidth of a disconnected graph is equal
to the maximum of the treewidths of its connected components.
A maximal graph with treewidth is called a -tree, while a graph with treewidth are known as partial -trees.
Graphs with treewidth
may be characterized by a finite set of forbidden
minors, as summarized in the following table. For the case of , more than 75 minimal forbidden minors of widely varying
structures are known (Sanders 1993, Sanders 1995, Chlebiková 2002).
octahedral graph ,
prism graph ,
unknown finite number of minors;
at least 75 known
The scramble number is the most powerful known
lower bound on the gonality of a graph and satisfies
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