Between O(nm) and O(nα)

This paper uses periodic matrix multiplication to improve the time complexities for a number of graph problems. The time for finding an asteroidal triple is reduced from O(nm) to O(n(2.82)), and the time for finding a star cutset, a two-pair, and a dominating pair is reduced from O(nm) to O(n(2.79)). It is also shown that each of these problems is at least as hard as one of three basic graph problems for which the best known algorithms run in times O(nm) and O(n(alpha)). We note that the fast matrix multiplication algorithms do not seem to be practical because of the enormous constants needed to achieve the asymptotic time bounds. These results are important theoretically for breaking the n(3) barrier rather than giving efficient algorithms for a user.