Self-stabilizing 2-minimal dominating set algorithms based on loop composition

Given a graph G = (V, E), a 2-minimal dominating set (2-MDS) of G is a minimal dominating set D subset of V such that D \ {p(i), p(j)} boolean OR {p(z)} is not a dominating set for any nodes p(1), p(j) is an element of D (p(i) not equal p(j)) and p(z) is not an element of D. We propose two silent self-stabilizing asynchronous distributed algorithms to find a 2-MDS. In both algorithms, we assume the weakly fair distributed daemon and that the processes have unique identifiers. The first one is for the general networks. The time complexity is O(nH) rounds, and the space complexity is O(A log n) bits per process, where n is the number of processes, H is the diameter of the network, and A is the maximum degree. The second one is for the networks of girth at least 7. The girth is the length of the shortest cycles in the network. The time complexity is O(nH) rounds, and the space complexity is O(log n) bits per process.