On the Cover Polynomial of a Digraph

There are many polynomials which can be associated with a graph G, the most well known perhaps being the Tutte polynomial T (G; x, y) (cf. [B74] or [T54]). In particular, for specific values of x and y, T (G; x, y) enumerates various features of G. For example, T (G; 1, 1) is just the number of spanning trees of G, T (G; 2, 0) is the number of acyclic orientations of G, T (G; 1, 2) is the number of connected subgraphs of G, etc. (see [JVW90] for details). However, for directed graphs, no analogue of the Tutte polynomial is known. In this paper we introduce the cover polynomial C(D; x, y) for a directed graph D and examine its relationships to other graph polynomials. While the cover polynomial is not exactly the directed analogue of the Tutte polynomial, it does have a number of properties which are comparable to those of T (G; x, y).