Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs

We study the parameterized complexity of three NP hard graph completion problems The MINIMUM FILL IN problem is to decide if a graph can be triangulated by adding at most k edges We develop an O k mn f k algorithm for this problem on a graph with n vertices and m edges In particular this implies that the problem is xed parameter tractable FPT The PROPER INTERVAL GRAPH COMPLETION problem motivated by molecular biology asks if a graph can be made proper interval by adding no more than k edges We show that the problem is FPT by providing a simple search tree based algorithm that solves it in linear time Similarly we show that the parameterized version of the STRONGLY CHORDAL GRAPH COMPLETION problem is FPT by giving an O m logn time algorithm for it All our algorithms can actually enumerate all possible k completions within the same time bounds AMS MOS subject classi cation Q R C