The vertex-cover polynomial of a graph Original Research Article

In this paper we define the vertex-cover polynomial Ψ(G,τ) for a graph G. The coefficient of τr in this polynomial is the number of vertex covers V′ of G with |V′|=r. We develop a method to calculate Ψ(G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1, 2,…,m} into the vertex set V(G) of a graph G, subject to f−1(x)∪f−1(y)≠∅ for every edge xy in G. Let F(G,m) be the number of such mappings f. We show that F(G,m) can be determined from Ψ(G,τ).