Point- and arc-reaching sets of vertices in a digraph

TL;DR

This paper extends the concepts of point- and arc-bases from finite to infinite digraphs, providing new insights into reachability sets in directed graphs.

Contribution

It generalizes existing finite digraph results to infinite digraphs, introducing new definitions and properties of point- and arc-reaching sets.

Findings

Extended point-bases results to infinite digraphs

Characterized minimal arc-reaching sets in infinite digraphs

Provided theoretical framework for reachability in infinite directed graphs

Abstract

In a digraph $D = (X, \mathcal{U})$, not necessarily finite, an arc $(x, y) \in \mathcal{U}$ is reachable from a vertex $u$ if there exists a directed walk $W$ that originates from $u$ and contains $(x, y)$. A subset $S \subseteq X$ is an arc-reaching set of $D$ if for every arc $(x, y)$ there exists a diwalk $W$ originating at a vertex $u \in S$ and containing $(x, y)$. A minimal arc-reaching set is an arc-basis. $S$ is a point-reaching set if for every vertex $v$ there exists a diwalk $W$ to $v$ originating at a vertex $u \in S$. A minimal point-reaching set is a point-basis. We extend the results of Harary, Norman, and Cartwright on point-bases in finite digraphs to point- and arc-bases in infinite digraphs.