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

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11914)

## 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 $$\mathcal {S}, \mathcal {T} \subset V$$.

## Keywords

Directed acyclic graph ST-reachable DAG Self-stabilization

## References

1. 1.
Chaudhuri, P., Thompson, H.: A self-stabilizing algorithm for the st-order problem. Int. J. Parallel Emergent Distrib. Syst. 23(3), 219–234 (2008)
2. 2.
Karaata, M.H., Chaudhuri, P.: A dynamic self-stabilizing algorithm for constructing a transport net. Computing 68(2), 143–161 (2002)
3. 3.
Kim, Y., Aono, H., Katayama, Y., Masuzawa, T.: A self-stabilizing algorithm for constructing a maximal (2,2)-directed acyclic mixed graph. In: the 6th International Symposium on Computing and Networking (CANDAR) (2018)Google Scholar
4. 4.
Kim, Y., Ohno, H., Katayama, Y., Masuzawa, T.: A self-stabilizing algorithm for constructing a maximal (1, 1)-directed acyclic mixed graph. Int. J. Netw. Comput. 8(1), 53–72 (2018)
5. 5.
Kim, Y., Shibata, M., Sudo, Y., Nakamura, J., Katayama, Y., Masuzawa, T.: A self-stabilizing algorithm for constructing an $$\cal{ST}$$-reachable directed acyclic graph when $$|\cal{S}| \le 2$$ and $$|\cal{T}| \le 2$$. In: Proceedings of the 39th IEEE International Conference on Distributed Computing Systems (ICDCS), pp. 2228–2237 (2019)Google Scholar

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Junya Nakamura
• 1
Email author
• Masahiro Shibata
• 2
• Yuichi Sudo
• 3
• Yonghwan Kim
• 4
1. 1.Toyohashi University of TechnologyToyohashiJapan
2. 2.Kyushu Institute of TechnologyIizukaJapan
3. 3.Osaka UniversitySuitaJapan
4. 4.Nagoya Institute of TechnologyNagoyaJapan