Tutte polynomials for oriented matroids

TL;DR: In this paper , a generalization of the Tutte polynomial to oriented graphs and regular oriented matroids is presented. But the A -polynomial invariant of an oriented graph is not equivalent to its regular oriented graph invariant.

Abstract: . The Tutte polynomial is a fundamental invariant of graphs and matroids. In this article, we define a generalization of the Tutte polynomial to oriented graphs and regular oriented matroids. To any regular oriented matroid N , we associate a polynomial invariant A N ( q, y, z ), which we call the A -polynomial . The A -polynomial has the following interesting properties among many others: We explore various properties and specializations of the A -polynomial. We show that some of the known properties or the Tutte polynomial of matroids can be extended to the A -polynomial of regular oriented matroids. For instance, we show that a specialization of A N counts all the acyclic orientations obtained by reorienting some elements of N , according to the number of reoriented elements.Letus mention that in a previous article we had defined an invariant of oriented graphs that we called the B -polynomial , which is also a generalization of the Tutte polynomial. However, the B -polynomial of an oriented graph N is not equivalent to its A -polynomial, and the B -polynomial cannot be extended to an invariant of regular oriented matroids.