  This paper studies the relation between node deletion and algebraic
connectivity for graphs with a hierarchical structure represented by layers. To
capture this structure, the concepts of layered path graph and its (sub)graph
cone are introduced. The problem is motivated by a mobile robot formation
control guided by a leader. In particular, we consider a scenario in which
robots may leave the network resulting in the removal of the nodes and the
associated edges. We show that the existence of at least one neighbor in the
upper layer is crucial for the algebraic connectivity not to deteriorate by
node deletion.
