Interval Partitions and Activities for the Greedoid Tutte Polynomial

Abstract

The two variable greedoid Tutte polynomialf(G;t,z), which was introduced in previous work of the authors, is studied via external activities. Two different partitions of the Boolean lattice of subsets are derived and a feasible set expansion off(G) is developed. All three of these results generalize theorems for matroids. One interval partition yields a characterization of antimatroids among the class of all greedoids. As an application, we prove that whenGis the directed branching greedoid associated with a rooted digraphD, then the highest power of (z+1) which dividesf(G) equals the minimum number of edges which can be removed fromDto produce an acyclic digraph in which all vertices ofDare still accessible from the root. The unifying theme behind these results is the idea of a computation tree.