Cut-Set Intersections and Node Partitions

Abstract

With any cut set of a graph can be associated a partition of the nodes into two cells. We show that each cut-set intersection term in the cut-set inclusion and exclusion formula can be associated with a k-cell partition of the nodes. Thus, the number of distinct terms in the cut-set inclusion and exclusion formula cannot exceed the number of partitions of the set of nodes. This leads to a simplified formula for graph reliability: the node partition formula, When finding the overall reliability in complete graphs no terms cancel, thus the number of terms is equal to the number of partitions. In other graphs we show that the number of non-cancelling terms in the cut-set inclusion and exclusion formula is equal to the number of minimal partition sets of the graph. It follows that cut-set inclusion and exclusion is inherently an inefficient method for the exact calculation of network reliability measures.