The vertex-cover polynomial of a graph

Abstract

In this paper we define the vertex-cover polynomial y(G,t) for a graph G. The coefficient of -f in this polynomial is the number of vertex covers V' of G with W'\ =r. We develop a method to calculate y(G, r). Motivated by a problem in biological systematics, we also consider the mappings / from {1, 2,...,m} into the vertex set V(G) of a graph G, subject to /~'(.v)U f-1)-40 for every ed8e xy in G- Let F(G,m) be the number of such mappings J. We show that F(G,m) can be determined from y(G,t). ©2002 Elsevier Science B.V. All rights reserved. ».