Complete Correspondence between the Quotient block topology and the Block-Cut Tree

Abstract

The block topology , generated by blocks of a connected graph , gives rise to a quotient space obtained by the identifying topologically indistinguishable vertices. In this paper,  our main result, the Hasse-block-cut correspondence, shows that the Hasse diagram of specialization poset of  is isomorphic to the block-cut tree , this holds under the condition that  has no internal bridge blocks (bridges between two cut vertices). This correspondence is a complete invariant as two such graphs have homeomorphic quotient spaces exactly when their block-cut trees are isomorphic, we also prove that any graph isomorphism induces a homeomorphism on the quotient spaces. In addition, we demonstrate that the degree of a node in is equal to the number of Hasse neighbors of the corresponding point in  . out results establish  can serve as a natural topological model for block decomposition, connecting graph theory and finite topology.