A Generalized Topology from the Edge Set of Maximal Paths of Directed Graphs

Abstract

This study intends to provide a fundamental step towards studying the properties of directed graphs with their corresponding generalized topological spaces. A generalized topology (GT) \(\mu\) on a nonempty set X is defined as a family of subsets of X such that \(\Theta\) and an arbitrary union of sets in \(\mu\) is in \(\mu\) . In this study, we introduce a new generalized topology generated by the set of edges of maximal paths of the directed graph D called the maximal path edge generalized topology (MPE\(\Gamma\)), denoted by \(\Gamma\)MP (D). The basic topological properties and connectedness in the context of this new structure are explored and illustrated. In particular, this paper established that (E(D); \(\Gamma\)MP (D)) is a strong generalized topological space and characterized the open and closed sets in this space. Moreover, it was seen that the MPE\(\Gamma\) space of every disconnected digraph is \(\Gamma\)MPMP -disconnected and the MPE\(\Gamma\) space of every connected digraph is also characterized.

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

“…There are many ways of associating topology from a graph, as seen in [1], [3], [5], [6], [7], and [8]. The most common method among these is by treating a collection of subsets of a nonempty set (e.g.…”