Some NP-Complete Problems Similar to Graph Isomorphism

The GRAPH ISOMORPHISM problem has so far resisted attempts at determining its complexity status—it has not been shown to be NP-complete nor in P. In this paper several altered or generalized versions of the ISOMORPHISM problem are presented and shown to be NP-complete. One of these is the problem of determining whether a given graph has a fixed-point-free automorphism. Some speculation is made on the possible implications of these results on deciding the complexity status of ISOMORPHISM. Various classes and hierarchies of problems in NP are discussed.