The vertex-cover polynomial of a graph

Authors:
F. M. Dong;M. D. Hendy;C. H. C. Little
Affiliations:
Institute of Fundamental Science (Mathematics), Massey University,Privat Bag 11222, Palmerston North, New Zealand;Institute of Fundamental Science (Mathematics), Massey University, Privat Bag 11222, Palmerston North, New Zealand;Institute of Fundamental Science (Mathematics), Massey University, Private Bag 11222, Palmerston North, New Zealand
Venue:
Discrete Mathematics
Year:
2002

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Abstract

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) ≠ φ 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).