On the complexity of computing treelength Original Research Article

We resolve the computational complexity of determining the treelength of a graph, thereby solving an open problem of Dourisboure and Gavoille, who introduced this parameter, and asked to determine the complexity of recognizing graphs of a bounded treelength Dourisboure and Gavoille (2007) . While recognizing graphs with treelength 1 is easily seen as equivalent to recognizing chordal graphs, which 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 a treelength at most image is NP-complete for every fixed image, and use this result to show that treelength in weighted graphs is hard to approximate within a factor smaller than image. Additionally, we show that treelength can be computed in time image by giving an exact exponential time algorithm for the Chordal Sandwich problem and showing how this algorithm can be used to compute the treelength of a graph.