Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Connectivity of the Lifts of a Greedoid. | Connectivity of the Lifts of a Greedoid Steven J. Tedford Department of Mathematics and Computer Science Franklin and Marshall College PO Box 3003 Lancaster PA 17604-3003 Submitted Jul 16 2006 Accepted Apr 30 2007 Published May 23 2007 Mathematics Subject Classification 05C99 05B99 Abstract 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 n-connected. Additionally for any cut-vertex v and cut-edge e of a graph r let C v be the component of r v containing the root and C e be the component of r 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 - land E C e e r - 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. 1 Introduction and Definitions The undirected branching greedoid of a rooted graph has as its feasible sets sets of edges whose induced subgraphs are trees which contain the root. In 3 Li Neumann-Lara and Rivera-Campo attempted to generalize this greedoid by defining the feasible sets of a new greedoid to be sets of edges whose induced graphs are connected contain the root are restricted in size and whose maximal sets span the graph. Although their construction does not always form a greedoid the iterated lift of the undirected branching greedoid has the properties that they were interested in. Since they only used the existence of a greedoid with these properties the existence of the iterated lift implies that the .
