Extended graph connectivity and its gradually increasing parallel complexity

Abstract

α-connectivity is a graph problem whose complexity gradually increases as the single parameter α grows. It is known that (i) α-connectivity is in NCt when α = c(log n)t-2/2, and (ii)it is P-complete when α = cnε. Unfortunately, the above (i) is only a result on increasing upper bounds. In this paper, we give more reasonable and stronger evidence that the complexity of α-connectivity really increases gradually as a grows. It is shown that α-connectivity can simulate gradually larger-size circuits as a grows. If at most α - 1 gates in each level of the circuit can have value 1, then α-connectivity can simulate gradually larger-depth circuits. We finally show evidence suggesting that these simulating powers of α-connectivity are best possible.