High Quality Content by WIKIPEDIA articles! In graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to speed up solving certain problems on the original graph. The treewidth measures the number of graph vertices mapped onto any tree node in an optimal tree decomposition. While it is NP-hard to determine the treewidth of a graph, many NP-hard combinatorial problems on graphs are solvable in polynomial time when restricted to graphs of bounded treewidth.