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 \(\mathcal {S}, \mathcal {T} \subset V\).