A self-stabilizing 2-minimal dominating set algorithm based on loop composition in networks of girth at least 7

We propose a silent self-stabilizing asynchronous distributed algorithm to find a 2-minimal dominating set (2-MDS) in networks of girth at least 7. Given a graph $G=(V, E)$, a 2-MDS of $G$ is a minimal dominating set $D\subseteq V$ such that $D\backslash \{p_{i},p_{j}\}\cup\{p_{z}\}$ is not a dominating set for any nodes $p_{i},p_{j}\in L (p_{i}\neq p_{j})$ and $p_{z}\ /{\!\!\!\in} D$. The girth is the length of the shortest cycles in the graph. We assume that the processes have unique identifiers. The proposed algorithm constructs a 2-MDS in the networks of girth at least 7 under the weakly fair distributed daemon. The time complexity is $O(nH)$ rounds, and the space complexity is $O(\log n)$ bits per process, where $n$ is the number of processes and $H$ is the diameter of the network.