A Worst-Case Time Upper Bound for Counting the Number of Independent Sets

The problem of counting the number of independent sets of a graph G (denoted as NI(G)) is a classic #P-complete problem for graphs of degree 3 or higher. Exploiting the strong relation between NI(G) and Fibonacci numbers, we show that if the depth-first graph of G does not contain a pair of basic cycles with common edges, then NI(G) can be computed in linear time (in the size of the graph). This determines new classes of instances of graphs without restrictions on their degrees and where the number of independent sets is computed in polynomial time. We design an exact deterministic algorithm for computing NI(G) based on the topological structure of the graph G, applying the wellknown splitting rule from Davis and Putnam (D&P) procedure. D&P is a familiar method for solving the Satisfiability Boolean Problem. Our algorithm for computing NI(G) establishes a leading Worst-Case Upper Bound of O(poly(n,m)∗1.220744), n and m being the number of nodes and edges of the graph G, respectively. The exact technique reported here can be used to compute the redundancy of a line in a communication network.