Connectivity of the Lifts of a Greedoid

Recently, attempts were made to generalize the undirected branching greedoid to a greedoid whose feasible sets consist of sets of edges containing the root satisfying additional size restrictions. Although this definition does not always result in a greedoid, the lift of the undirected branching greedoid has the properties desired by the authors. The k-th lift of a greedoid has sets whose nullity is at most k in the original greedoid. We prove that if the greedoid is n-connected, then its lift is also nconnected. Additionally, for any cut-vertex v and cut-edge e of a graph Γ, let C(v) be the component of Γ \ v containing the root and C(e) be the component of Γ \ e containing the root. We prove that if the k-th lift of the undirected branching greedoid is 2-connected, then |E(C(v))| < |V (C(v))| + k − 1 and |E(C(e))| > |E(Γ)| − k − 2. We also give examples indicating that no sufficient conditions for the kth lift to be 2-connected exists similar to these necessary conditions.