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. The treewidth of a disconnected graph is equal to the maximum of the treewidths of its connected components. A maximal graph with treewidth k is called a k-tree, while a graph with treewidth <=k are known as partial k-trees.
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