The Block Topological Space and Block Topological Graph Induced by Undirected Graphs

Abstract

Let $G=(V(G),E(G))$ be a simple undirected graph. A \textit{block} of $G$ is a maximal connected subgraph of $G$ that contains no cut-vertices \cite{eric}. The family of vertex sets of blocks of $G$ generates a unique topology. In this paper, we formally define the topology generated by the family of blocks in a graph called the \textit{block topological space}. Moreover, we characterize and describe some special attributes of the block topological space. Finally, we associate a corresponding graph from a given block topological space by defining the \textit{block topological graph}.