The homology of the cycle matroid of a coned graph

The cone G ˆ of a finite graph G is obtained by adding a new vertex p , called the cone point, and joining each vertex of G to p by a simple edge. We show that the rank of the reduced homology of the independent set complex of the cycle matroid of G ˆ is the cardinality of the set of the edge-rooted forests in the base graph G . We also show that there is a basis for this homology group such that the action of the automorphism group Aut ( G ) on this homology is isomorphic (up to sign) to that on the set of the edge-rooted forests in G .