Graph Isomorphism is in the Low Hierarchy

The problem of determining whether two given finite graphs are isomorphic is easily seen to belong to the class NP. But up to now, no polynomial time algorithm is known. On the other hand, no NP-completeness proof is known either. That is, GRAPH ISOMORPHISM is one of the few “open problems” in NP according to Garey and Johnson’s terminology [9]. Note that GRAPH ISOMORPHISM is already mentioned in Karp’s seminal paper [15]. Any attempt to prove NP-completeness of GRAPH ISOMORPHISM was frustrated by the apparently much more constrained nature of this problem as compared with the usual NP-complete problems. This led on the one hand to the conjecture that GRAPH ISOMORPHISM is not NP-complete (possibly in P); on the other hand that it might be NP-complete w.r.t. some weaker kind of reducibility (like y-reducibility, see open problem 1 in [ 1 I). Since Ladner’s work [ 181 we know that there exist problems in NP neither being NP-complete nor being in P, assuming P # NP. Hence, GRAPH ISOMORPHISM might very well be one of these “intermediate” problems. It seems that there is more evidence that GRAPH ISOMORPHISM is in P (or at least “close” to P) than for NP-completeness. Indeed, some (nontrivial) special cases of the graph isomorphism problem have been shown to be in P and also some in NP n co-NP (cf. [ 141). Further, unlike the known NP-complete problems, the corresponding counting problem (given two graphs, compute the number of different isomorphisms between them) has been shown to be of the same “degree” of complexity as the mere decision problem [22, see also 141. Note that the counting versions of many NP-complete problems are # P-complete, but # P is not even known to be included in the polynomial hierarchy [Z]. Very recently, it has been observed that graph (non-) isomorphism can be 312 0022~0000/88 53.00