Brief Announcement: Self-stabilizing Construction of a Minimal Weakly $\mathcal {ST}$-Reachable Directed Acyclic Graph

Abstract

In this paper, we propose a self-stabilizing algorithm to construct a minimal weakly $\mathcal {ST}$-reachable directed acyclic graph (DAG). Given an arbitrary simple, connected, and undirected graph $G=(V, E)$ and two sets of vertices, senders $\mathcal {S} (\subset V)$ and targets $\mathcal {T} (\subset V)$, a directed subgraph $\overrightarrow{G}$ of G is a weakly $\mathcal {ST}$-reachable DAG on G if $\overrightarrow{G}$ is a DAG and every sender can reach at least one target, and every target is reachable from at least one sender in $\overrightarrow{G}$. We say that a weakly $\mathcal {ST}$-reachable DAG $\overrightarrow{G}$ on G is minimal if any proper subgraph of $\overrightarrow{G}$ is no longer a weakly $\mathcal {ST}$-reachable DAG. The weakly $\mathcal {ST}$-reachable DAG on G, which we consider here, is a relaxed version of the original (or strongly) $\mathcal {ST}$-reachable DAG on G where all targets are reachable from all senders. A strongly $\mathcal {ST}$-reachable DAG G does not always exist; even if we focus on the case $|\mathcal {S}|=|\mathcal {T}|=2$, some G has no strongly $\mathcal {ST}$-reachable DAG. On the other hand, the proposed algorithm always construct a weakly $\mathcal {ST}$-reachable DAG for any given graph $G=(V, E)$ and any $\mathcal {S}, \mathcal {T} \subset V$.