On the complexity of testing for odd holes and induced odd paths

Abstract The following problems, of possible interest with regards to perfect graphs, are shown to be NP-Complete. 1. (1) Does a graph contain an induced odd cycle of length greater than three, passing through a prescribed vertex? 2. (2) Does a graph contain an induced odd path between two prescribed vertices? 3. (3) Does a graph contain an induced odd path between every two vertices?