Determining treelength is hard Daniel

In this article we resolve the computational complexity of determining the treelength of a graph, thereby solving an open problem of Dourusboure and Gavoille, who introduced this parameter, and asked to determine the complexity of recognizing graphs of bounded treelength [7]. The treelength of a graph G deals with the length of a tree-decomposition. The length of a tree decomposition of G is the largest distance in G between any two vertices appearing in the same tree node. The treelength of G is the minimum length of a tree decomposition of G. While recognizing graphs with treelength 1 is easily seen as equivalent to recognizing chordal graphs, something that can be done in linear time, the computational complexity of recognizing graphs with treelength 2 was unknown until this result. We show that the problem of determining whether a given graph has treelength at most k is NP-complete for every fixed k ≥ 2.