An algorithm for calculating the independence and vertex-cover polynomials of a graph

Abstract

The independence polynomial, ω ( G , x ) = ∑ w k x k , of a graph, G, has coefficients, w k , that enumerate the ways of selecting k vertices from G so that no two selected vertices share an edge. The independence number of G is the largest value of k for which w k ≠ 0 . Little is known of less straightforward relationships between graph structure and the properties of ω ( G , x ) , in part because of the difficulty of calculating values of w k for specific graphs. This study presents a new algorithm for these calculations which is both faster than existing ones and easily adaptable to high-level computer languages.