Tractable and Intractable Problems on Generalized Chordal Graphs

A generalized chordal graph is characterized by some positive integer k 3, and whose each cycle of length greater than k has a chord. Several tractable and intractable problems on a k-chordal graph are considered. When k is a xed positive integer, the recognition problem for k-chordalness is in NC. On a k-chordal graph, the maximal acyclic set problem is in NC when k = 3, and the problem is in RNC for any xed k > 3. Next we show that the recognition problem for k-chordalness is coNP-complete for k = (n), where n is the number of vertices. We also show that for any positive constant , several NP-complete problems on a general graph is also NP-complete on a k-chordal graph even if k = (n ). That concludes that the recognition problem for k-chordalness is coNP-complete even for k = (n ) for any positive constant .