Is Hyper-extensionality Preservable Under Deletions of Graph Elements?

Any hereditarily finite set S can be represented as a finite pointed graph –dubbed membership graph– whose nodes denote elements of the transitive closure of {S} and whose edges model the membership relation. Membership graphs must be hyper-extensional –nodes are pairwise not bisimilar– and bisimilar nodes represent the same hereditarily finite set. It is worth to notice that the removal of even a single node or edge from a membership graph can cause “collapses” of different nodes and, therefore, the loss of hyper-extensionality of the graph itself. With the intent of gaining a deeper understanding of the class of hereditarily finite sets, this paper investigates whether pointed hyper-extensional graphs always contain either a node or an edge whose removal does not disrupt the hyper-extensionality property.