On the Complexity of Computing Treelength

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 bounded treelength [6]. 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 treelength at most k is NP-complete for every fixed k ≥ 2, and use this result to show that treelength in weighted graphs is hard to approximate within a factor smaller than 3 2 . Additionally, we show that treelength can be computed in time O∗

• Type: Conference
• Year: 2007
• Where: MFCS
• Authors: Daniel Lokshtanov
• Keywords: Chordal Graphs, Computational Complexity, MFCS 2007, Open Problem