Computing transparently: the independent sets in a graph

Abstract

A procedure is given for finding the independent sets in an undirected graph by xeroxing onto transparent plastic sheets. Let an undirected graph having n vertices and m edges be given. A list of all the independent subsets of the set of vertices of the graph is constructed by using a xerox machine in a manner that requires the formation of only n + m + 1 successive transparencies. An accompanying list of the counts of the elements in each independent set is then constructed using only O(n ²) additional transparencies. The list with counts provides a list of all maximum independent sets. This gives an O(n ²) step solution for the classical problem of finding the cardinality of a maximal independent set in a graph. The applicability of these procedures is limited, of course, by the increase in the information density on the transparencies when n is large. Our ultimate purpose here is to give hand tested ‘ultra parallel’ algorithmic procedures that may prove suitable for realization using future optical technologies.